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Gottlob Frege (b. 1848, d. 1925) was a German mathematician, = logician, and=20 philosopher who worked at the University of Jena. Frege essentially = reconceived=20 the discipline of logic by constructing a formal system which, in = effect,=20 constituted the first =91predicate calculus=92. In this formal system, = Frege=20 developed an analysis of quantified statements and formalized the notion = of a=20 =91proof=92 in terms that are still accepted today. Frege then = demonstrated that one=20 could use his system to resolve theoretical mathematical statements in = terms of=20 simpler logical and mathematical notions. One of the axioms that Frege = later=20 added to his system, in the attempt to derive significant parts of = mathematics=20 from logic, proved to be inconsistent. Nevertheless, his definitions (of = the=20 predecessor relation and of the concept of natural = number) and=20 methods (for deriving the axioms of number theory) constituted a = significant=20 advance. To ground his views about the relationship of logic and = mathematics,=20 Frege conceived a comprehensive philosophy of language that many = philosophers=20 still find insightful. However, his lifelong project, of showing that=20 mathematics was reducible to logic, was not successful.
Frege founded the modern discipline of logic by developing a superior = method=20 of formally representing the logic of thoughts and inferences. He did = this by=20 developing: (a) a formal system which formed the basis of modern logic, = (b) an=20 elegant analysis of complex sentences and quantifier phrases that showed = an=20 underlying unity to certain classes of inferences, (c) a deep = understanding of=20 proof and definition, (d) a theory of extensions = which, though=20 seriously flawed, offered an intriguing picture of the foundations of=20 mathematics, (e) an insightful analysis of statements about number = (i.e., of=20 answers to the question =91How many?=92), (f) definitions and proofs of = some of the=20 basic axioms of number theory from a limited set of logically primitive = concepts=20 and axioms, and (g) a conception of logic as a discipline which has some = compelling features. We discuss these developments in the following=20 subsections.
In an attempt to realize Leibniz's ideas for a language of thought = and a=20 rational calculus, Frege developed a formal notation for regimenting = thought and=20 reasoning. Though this notation was first outlined in his=20 Begriffsschrift (1879), the most mature statement of Frege's = system was=20 in his 2volume Grundgesetze der Arithmetik (1893/1903). = Frege's=20 1893/1903 system is best characterized as a logic of terms which, with = the help=20 of a few definitions, grounds the modern predicate calculus. A predicate = calculus is a formal system (a formal language and a method of proof) in = which=20 one can represent valid inferences among predications, i.e., among = statements in=20 which properties are predicated of objects. Frege's earlier 1879 system = was more=20 of a predicate calculus, and as such, was the first of its kind.
In this subsection, we shall examine the most basic elements of = Frege's=20 1893/1903 term logic and predicate calculus. These are the statements = involving=20 function applications and the simple predications which fall out as a = special=20 case.
In Frege's term logic, the complete expressions are all terms, i.e., = denoting=20 expressions. These include: (a) simple names of objects, like =912=92 = and =91π=92, (b)=20 complex terms which denote objects, like =912^{2}=92 and =913 + = 1=92, and (c)=20 sentences (which are also complex terms). The complex terms in (b) and = (c) are=20 formed with the help of =91incomplete expressions=92 which signify = functions, such=20 as the unary squaring function =91( )^{2}=92 and the binary = addition=20 function =91( )+( )=92. In these functional expressions, = =91( )=92 is=20 used as a placeholder for what Frege called the arguments of = the=20 function; the placeholder reveals that the expressions signifying = function are,=20 on Frege's view, incomplete and stand in contrast to complete = expressions such=20 as those in (a), (b), and (c). (Though Frege thought it inappropriate to = call=20 the incomplete expressions that signify functions =91names=92, we shall = sometimes do=20 so in what follows, though the reader should be warned that Frege had = reasons=20 for not following this practice.) Thus, a mathematical expression such = as=20 =912^{2}=92 denotes the result of applying the function = ( )^{2}=20 to the number 2 as argument, namely, the number 4. Similarly, the = expression =917=20 + 1=92 denotes the result of applying the binary function = +(( ),( )) to=20 the numbers 7 and 1 as arguments, in that order.
Even the sentences of Frege's mature logical system are complex = terms; they=20 are terms that denote truthvalues. Frege distinguished two=20 truthvalues, The True and The False, which he took to be objects. The = basic=20 sentences of Frege's system are constructed using the expression=20 =91( ) =3D ( )=92, which signifies a binary function = that maps a=20 pair of objects x and y to The True if x is = identical=20 to y and maps x and y to The False otherwise. = A=20 sentence such as =912^{2} =3D 4=92 therefore denotes the = truthvalue The True,=20 while the sentence =912^{2} =3D 6=92 denotes the truthvalue The = False.
An important class of these identity statements are statements of the = form=20 =91=83(x) =3D y=92, where =83( ) is any unary = function (i.e.,=20 function of a single variable), x is the argument of the = function, and=20 =83(x) is the value that the function =83( ) has for the = argument=20 x. There can also be identity statements involving binary = functions=20 (i.e., functions of two variables), namely, =83(x,y) = =3D=20 z. And so on, for functions of more than two variables.
If we replace a complete name appearing in a sentence by a = placeholder, the=20 result is an incomplete expression that signifies a special kind of = function=20 which Frege called a concept. Concepts are functions = which map=20 every argument to one of the truthvalues. Thus, =91( )>2=92 = denotes the=20 concept being greater than 2, which maps every object greater = than 2 to=20 The True and maps every other object to The False. Similarly,=20 =91( )^{2} =3D 4=92 denotes the concept that which when = squared is=20 identical to 4. Frege would say that any object that a concept maps = to The=20 True falls under the concept. Thus, the number 2 falls under = the=20 concept that which when squared is identical to 4. In what = follows, we=20 use lowercase expressions like =83( ) to talk generally about = functions, and=20 uppercase expressions like F( ) to talk more specifically = about=20 those functions which are concepts.
Frege supposed that a mathematical claim such as =912 is prime=92 = should be=20 formally represented as =91P(2)=92. The verb phrase =91is = prime=92 is thereby=20 analyzed as denoting the concept P( ) which maps primes to = The=20 True and everything else to The False. Thus, a simple = predication like=20 =912 is prime=92 becomes analyzed in Frege's system as a special case of = functional=20 application.
The preceding analysis of simple mathematical predications led Frege = to=20 extend the applicability of this system to the representation of=20 nonmathematical thoughts and predications. This move formed the basis = of the=20 modern predicate calculus. Frege analyzed a nonmathematical predicate = like =91is=20 happy=92 as signifying a function of one variable which maps its = arguments to a=20 truthvalue. Thus, =91is happy=92 denotes a concept which can be = represented in the=20 formal system as =91H( )=92. H( ) maps those = arguments=20 which are happy to The True, and maps everything else to The False. The = sentence=20 =91John is happy=92 (=91H(j)=92) is thereby analyzed = as: the object=20 denoted by =91John=92 falls under the concept signified by =91( ) = is happy=92.=20 Thus, a simple predication is analyzed in terms of falling under a = concept,=20 which in turn, is analyzed in terms of functions which map their = arguments to=20 truth values. By contrast, in the modern predicate calculus, this last = step of=20 analyzing predication in terms of functions is not assumed; predication = is seen=20 as more fundamental than functional application. The sentence =91John is = happy=92 is=20 formally represented as =91Hj=92, where this is a basic form of = predication=20 (=91the object j instantiates or exemplifies the property = H=92).=20 In the modern predicate calculus, functional application is analyzable = in terms=20 of predication, as we shall soon see.
In Frege's analysis, the verb phrase =91loves=92 signifies a binary = function of=20 two variables: L(( ),( )). This function takes a pair = of=20 arguments x and y and maps them to The True if = x=20 loves y and maps all other pairs of arguments to The False. = Although it=20 is a descendent of Frege's system, the modern predicate calculus = analyzes=20 loves as a twoplace relation (Lxy) rather than a = function;=20 some objects stand in the relation and others do not. The difference = between=20 Frege's understanding of predication and the one manifest by the modern=20 predicate calculus is simply this: in the modern predicate calculus, = relations=20 are taken as basic, and functions are defined as a special case of = relation,=20 namely, those relations R such that for any objects x, = y, and z, if Rxy and Rxz, then = y=3Dz.=20 By contrast, Frege took functions to be more basic than relations. His = logic is=20 based on functional application rather than predication, and so = relations become=20 analyzed as a binary functions which map a pair of arguments to a = truthvalue.=20 Thus, a 3place relation like gives would be analyzed in = Frege's logic=20 as a function that maps arguments x, y, and z = to an=20 appropriate truthvalue depending on whether x gives y = to=20 z; the 4place relation buys would be analyzed as a = function=20 that maps the arguments x, y, z, and = u to an=20 appropriate truthvalue depending on whether x buys y = from=20 z for amount u; etc.
So far, we have been discussing Frege's analysis of =91atomic=92 = statements. To=20 complete the basic logical representation of thoughts, Frege added = notation for=20 representing more complex statements (such as negated and conditional=20 statements) and statements of generality (those involving the = expressions=20 =91every=92 and =91some=92). Though we no longer use his notation for = representing=20 complex and general statements, it is important to see how the notation = in=20 Frege's term logic already contained all the expressive power of the = modern=20 predicate calculus.
There are four special functional expressions which are used in = Frege's=20 system to express complex and general statements:
Intuitive
SignificanceFunctional Expression The Function It = Signifies Statement The function which maps The True to The True and maps all = other=20 objects to The False; this is used to indicate that the argument = is a=20 true statement that names The True. Negation The function which maps The True to The False and maps all = other=20 objects to The True Conditional The function which maps a pair of objects to The False if the = first=20 (i.e., the one named in the bottom branch) is The True and the = second=20 isn't The True, and maps all other pairs of objects to The = True Generality The secondlevel function which maps a firstlevel concept=20 F( ) to The True if F( ) maps every = object=20 to The True; otherwise it maps F( ) to The=20 False.
The best way to understand this notation is by way of some tables, = which show=20 some specific examples of statements and how those are rendered in = Frege=92=20 notation and in the modern predicate calculus.
The first table shows how Frege's logic can express the = truthfunctional=20 connectives such as not, ifthen, and, or, and ifandonlyif.
Example Frege's Notation Modern Notation John is happy Hj It is not the case that John is happy =ACHj If the sun is shining, then John is happy = Ss & Hj Either the sun is shining or John is happy <= /TD> Ss =20 Hj The sun is shining if and only if John is = happy = Ss ≡=20 Hj
As one can see, Frege didn't use the primitive connectives =91and=92, = =91or=92, or=20 =91if and only if=92, but always used canonical equivalent forms defined = in terms of=20 negations and conditionals. Note the last row of the table =97 when = Frege wants to=20 assert that two conditions are materially equivalent, he uses the = identity sign,=20 since this says that they denote the same truthvalue. In the modern = sentential=20 calculus, the biconditional does something equivalent, for a statement = of the=20 form φ≡ψ is true whenever φ and ψ are both = true or both false. The only=20 difference is, in the modern sentential calculus φ and ψ are = not construed as=20 terms denoting truthvalues, but rather as sentences having truth = conditions=20 (though, in the semantics of the sentential calculus, sentences are = assigned=20 truthvalues as their =91semantic value=92, and they are considered = true/false=20 according to which truthvalue serves as their semantic value).
The table below compares statements of generality in Fregestyle = notation and=20 in the modern predicate calculus. We say =91Fregestyle" notation = because we are=20 modifying Frege's notation a bit so as to simplify the presentation; we = shall=20 not use the special typeface (Gothic) that Frege used for variables in = general=20 statements, or observe some of the special conventions that he adopted, = for=20 reasons that would distract us from this introduction.
=
Example FregeStyle Notation Modern Notation Everything is mortal ∀xMx Something is mortal =AC∀x=ACMx
i.e.,=20 ∃xMxNothing is mortal ∀x=ACMx
i.e., = =AC∃xMxEvery person is mortal ∀x(Px → = Mx) Some person is mortal <= /TD> =AC∀x(Px → = =ACMx)
i.e.,=20 ∃x(Px & Mx)No person is mortal ∀x(Px → = =ACMx)
i.e.,=20 =AC∃x(Px & Mx)All and only persons are mortal = ∀x(Px ≡=20 Mx)
Note the last line. Here again, Frege uses the identity sign to help = state=20 the material equivalence of two concepts. He can do this because = materially=20 equivalent concepts F and G are such that F = maps an=20 object x to The True whenever G maps x to The = True;=20 i.e., for all arguments x, F and G map = x to=20 the same truthvalue.
In the modern predicate calculus, the symbols =91∀=92 = (=91every=92) and =91∃=92 (=91some=92)=20 are called the =91universal=92 and =91existential=92 quantifier, = respectively, and the=20 variable =91x=92 in the sentence =91∀xMx=92 is = called a =91quantified=20 variable=92, or =91variable bound by the quantifier=92. We will follow = this practice=20 of calling statements involving one of these quantifier phrases = =91quantified=20 statements=92. As one can see from the table above, Frege didn't use an=20 existential quantifier. He was aware that a statement of the form=20 =91∃x(=85)=92 could always be defined as = =91=AC∀x=AC(=85)=92.
It is important to mention here that the predicate calculus = formulable in=20 Frege's logic is a =91secondorder=92 predicate calculus. This means it = allows=20 quantification over functions as well as quantification over objects; = i.e.,=20 statements of the form =91Every function =83 is such that =85=92 and = =91Some function =83 is=20 such that =85=92 are allowed. Thus, the statement =91objects a = and b=20 fall under the same concepts=92 would be written as follows in = Fregestyle=20 notation:
=
whereas in the modern secondorder predicate calculus, we write this = as:
∀F(Fa ≡ = Fb)
In what follows, we shall continue to use the notation of the modern=20 predicate calculus instead of Fregestyle notation. In particular, we = adopt the=20 following conventions. (1) We shall often use =91Fx=92 instead = of=20 =91F(x)=92 to represent the fact that x falls = under the=20 concept F; we use =91Rxy=92 instead of=20 =91R(x,y)=92 to represent the fact that = x stands=20 in the relation R to y; etc. (2) Instead of using = expressions=20 with placeholders, such as =91( ) =3D ( )=92 and=20 =91P( )=92, to signify functions and concepts, we shall = simply use =91=3D=92=20 and =91P=92. (3) When replace one of the complete names in a = sentence by a=20 variable, the resulting expression will be called an open = sentence or=20 an open formula. Thus, whereas =913<2=92 is a sentence,=20 =913<x=92 is an open sentence; and whereas =91Hj=92 = is a formal=20 sentence that might be used to represent =91John is happy=92, the = expression=20 =91Hx=92 is an open formula which might be rendered = =91x is happy=92=20 in natural language. (4) Finally, we shall on occasion employ the Greek = symbol φ=20 as a metavariable ranging over formal sentences, which may or may not be = open.=20 Thus, =91φ(a)=92 will be used to indicate any sentence = (simple or complex)=20 in which the name =91a=92 appears; =91φ(a)=92 is = not to be=20 understood as Fregenotation for a function φ applied to argument = a.=20 Similarly, =91φ(x)=92 will be used to indicate an open = sentence in which=20 the variable x may or may not be free, not a function of=20 x.
Frege's functional analysis of predication coupled with his = understanding of=20 generality freed him from the limitations of the =91subjectpredicate=92 = analysis of=20 ordinary language sentences that formed the basis of Aristotelian logic = and it=20 made it possible for him to develop a more general treatment of = inferences=20 involving =91every=92 and =91some=92. In traditional Aristotelian logic, = the subject of=20 a sentence and the direct object of a verb are not on a logical par. The = rules=20 governing the inferences between statements with different but related = subject=20 terms are different from the rules governing the inferences between = statements=20 with different but related verb complements. For example, in = Aristotelian logic,=20 the rule which permits the valid inference from =91John loves Mary=92 to = =91Something=20 loves Mary=92 is different from the rule which permits the valid = inference from=20 =91John loves Mary=92 to =91John loves something=92. The rule governing = the first=20 inference is a rule which applies only to subject terms whereas the rule = governing the second inference governs reasoning within the predicate, = and thus=20 applies only to the transitive verb complements (i.e., direct objects). = In=20 Aristotelian logic, these inferences have nothing in common.
In Frege's logic, however, a single rule governs both the inference = from=20 =91John loves Mary=92 to =91Something loves Mary=92 and the inference = from =91John loves=20 Mary=92 to =91John loves something=92. That's because the subject John = and the direct=20 object Mary are both considered on a logical par, as arguments of the = function=20 loves. In effect, Frege saw no logical difference between the = subject=20 =91John=92 and the direct object =91Mary=92. What is logically important = is that =91loves=92=20 denotes a function of 2 arguments. No matter whether the quantified = expression=20 =91something=92 appears as subject (=91Something loves Mary=92) or = within a predicate=20 (=91John loves something=92), it is to be resolved in the same way. In = effect, Frege=20 treated these quantified expressions as variablebinding operators. The=20 variablebinding operator =91some x is such that=92 can bind = the variable=20 =91x=92 in the open sentence =91x loves Mary=92 as = well as the=20 variable =91x=92 in the open sentence =91John loves = x=92. Thus, Frege=20 analyzed the above inferences in the following general way:
Both inferences are instances of a single valid inference rule. To = see this=20 more clearly, here are the formal representations of the above informal=20 arguments:
The logical axiom which licenses both inferences has the form:
Ra_{1}=85a_{i}=85<= EM>=20 a_{n} →=20 ∃x(Ra_{1}=85x=85 = a_{n}),
where R is a relation that can take n arguments, = and=20 a_{1},=85,a_{n} are any = constants=20 (names), for any a_{i} such that=20 1≤i≤n. This logical axiom tells us that = from a simple=20 predication involving an nplace relation, one can = existentially=20 generalize on any argument, and validly derive a existential statement. =
Indeed, this axiom can be made even more general. If = φ(a) is any=20 statement (formula) in which a constant (name) a appears, and=20 φ(x) is the result of replacing one or more occurrences of = a=20 by x, then the following is a logical axiom:
φ(a) → = ∃xφ(x)
The inferences which start with the premise =91John loves Mary=92, = displayed=20 above, both appeal to this axiom for justification. This axiom is = actually=20 derivable as a theorem from Frege's Basic Law IIa (1893, =A747). Basic = Law IIa=20 asserts ∀xφ(x) → φ(a), = and the above axiom for the=20 existential quantifier can be derived from IIa using the rules governing = conditionals, negation, and the definition of ∃x(=85) = discussed above.=20
There is one other consequence of Frege's logic of quantification = that should=20 be mentioned. Frege took claims of the form ∃x(=85) to be = existence=20 claims. He suggested that existence is not a concept under = which=20 objects fall but rather a secondlevel concept under which firstlevel = concepts=20 fall. A concept F falls under this secondlevel concept just in = case=20 F maps at least one object to The True. So the claim = =91Martians don't=20 exist=92 is analyzed as an assertion about the concept martian, = namely,=20 that nothing falls under it. Frege therefore took existence to = be that=20 secondlevel concept which maps a firstlevel concept F to The = True=20 just in case ∃xFx and maps all other concepts to The = False. Many=20 philosophers have thought that this analysis validates Kant's view that=20 existence is not a (real) predicate.
Frege's system (i.e., his term logic/predicate calculus) consisted of = a=20 language and an apparatus for proving statements. The latter consisted = of a set=20 of logical axioms (statements considered to be truths of logic) and a = set of=20 rules of inference that lay out the conditions under which certain = statements of=20 the language may be correctly inferred from others. Frege made a point = of=20 showing how every step in a proof of a proposition was justified either = in terms=20 of one of the axioms or in terms of one of the rules of inference or = justified=20 by a theorem or derived rule that had already been proved.
Thus, as part of his formal system, Frege developed a strict = understanding of=20 a =91proof=92. In essence, he defined a proof to be any finite sequence = of=20 statements such that each statement in the sequence either is an axiom = or=20 follows from previous members by a valid rule of inference. Thus, a = proof of a=20 theorem of logic, say φ, is therefore any finite sequence of = statements (with φ=20 the final statement in the sequence) such that each member of the = sequence: (a)=20 is one of the logical axioms of the formal system, or (b) follows from = previous=20 members of the sequence by a rule of inference. These are essentially = the=20 definitions that logicians still use today.
Frege was extremely careful about the proper description and = definition of=20 logical and mathematical concepts. He developed powerful and insightful=20 criticisms of mathematical work which did not meet his standards for = clarity.=20 For example, he criticized mathematicians who defined a variable to be a = number=20 that varies rather than an expression of language which can vary as to = which=20 determinate number it refers to. And he criticized those mathematicians = who=20 developed =91piecemeal=92 definitions or =91creative=92 definitions. In = the=20 Grundgesetze der Arithmetik, II (1903, Sections 5667) Frege = criticized=20 the practice of defining a concept on a given range of objects and later = redefining the concept on a wider, more inclusive range of concepts. = Frequently,=20 this =91piecemeal=92 style of definition led to conflict, since the = redefined=20 concept did not always reduce to the original concept when one restricts = the=20 range to the original class of objects. In that same work (1903, = Sections=20 139147), Frege criticized the mathematical practise of introducing = notation to=20 name (unique) entities without first proving that there exist (unique) = such=20 entities. He pointed out that such =91creative definitions=92 were = simply=20 unjustified.
Frege's ontology consisted of two fundamentally different types of = entities,=20 namely, functions and objects (1891, 1892b, 1904). Functions are in some = sense=20 =91unsaturated=92; i.e., they are the kind of thing which take objects = as arguments=20 and map those arguments to a value. This distinguishes them from = objects. As=20 we've seen, the domain of objects included two special objects, namely, = the=20 truthvalues The True and The False.
In his work of 1893/1903, Frege attempted to expand the domain of = objects by=20 systematically associating, with each function =83, an object which he = called=20 the courseofvalues of =83. The courseofvalues of a function = is a=20 record of the value of the function for each argument. The principle = Frege used=20 to systematize coursesofvalues is Basic Law V (1893/=A720;):
The courseofvalues of the concept =83 is identical to the=20 courseofvalues of the concept g if and only if =83 and = g=20 agree on the value of every argument (i.e., if and only if for every = object=20 x, =83(x) =3D g(x)).
Frege used the following notation to denote the courseofvalues of = the=20 function =83:
<= /BLOCKQUOTE>where the first occurrence of the Greek ε (with the = smoothbreathing mark=20 over it) is a =91variablebinding operator=92 which we might read as = =91the=20 courseofvalues of=92. Using this notation, Frege formally represented = Basic Law=20 V in his system as:
Basic Law V
(Actually, Frege used an identity sign instead of the biconditional = as the=20 main connective of this principle, for reasons described above.)
Frege called the courseofvalues of a concept F its=20 extension. The extension of a concept F records just = those=20 objects which F maps to The True. Thus Basic Law V applies = equally well=20 to the extensions of concepts. Let =91φ(x)=92 be an open = sentence of any=20 complexity with the free variable x (the variable x = may have=20 more than one occurrence in φ(x), but for simplicity, = assume it has=20 only one occurrence). Then using the variablebinding operator ε, = Frege would=20 use the expression =91εφ(ε)=92 (with a smoothbreathing = mark over the first epsilon=20 and where the second epsilon replaces x in φ(x)) = to denote the=20 extension of the concept φ. Where =91n=92 is the name of = an object, Frege=20 could define =91object n is an element of the extension of the = concept φ=92=20 in the following simple terms: =91the concept φ maps n to = The True=92=20 (i.e., φ(n)). For example, the number 3 is an element of = the extension=20 of the concept odd number greater than 2 if and only if this = concept=20 maps 3 to The True.
Unfortunately, Basic Law V implies a contradiction, and this was = pointed out=20 to Frege by Bertrand Russell just as the second volume of the=20 Grundgesetze was going to press. Russell recognized that some=20 extensions are elements of themselves and some are not; the extension of = the=20 concept extension is an element of itself, since that concept = would map=20 its own extension to The True. The extension of the concept = spoon is=20 not an element of itself, because that concept would map its own = extension to=20 The False (since extensions aren't spoons). But now what about the = concept=20 extension which is not an element of itself? Let E = represent=20 this concept and let e name the extension of E. Is = e=20 an element of itself? Well, e is an element of itself if and = only if=20 E maps e to The True (by the definition of =91element = of=92 given=20 at the end of the previous paragraph, where e is the extension = of the=20 concept E). But E maps e to The True if and = only if=20 e is an extension which is not an element of itself, i.e., if = and only=20 if e is not an element of itself. We have thus reasoned that = e=20 is an element of itself if and only if it is not, showing the = incoherency in=20 Frege's conception of an extension.
Further discussion of this problem can be found in the entry on Russell's = Paradox,=20 and a more complete explanation of how the paradox arises in Frege's = system is=20 presented in the entry on Frege's logic, = theorem,=20 and foundations for mathematics.
2.4.2 Proposed Foundation for Mathematics
Before he became aware of Russell's paradox, Frege attempted to = construct a=20 logical foundation for mathematics. Using the logical system containing = Basic=20 Law V (1893/1903), he attempted to demonstrate the truth of the = philosophical=20 thesis known as logicism, i.e., the idea not only that = mathematical=20 concepts can be defined in terms of purely logical concepts but also = that=20 mathematical principles can be derived from the laws of logic alone. But = given=20 that the crucial definitions of mathematical concepts were stated in = terms of=20 extensions, the inconsistency in Basic Law V undermined Frege's attempt = to=20 establish the thesis of logicism. Few philosophers today believe that=20 mathematics can be reduced to logic in the way Frege had in mind. = Mathematical=20 theories such as set theory seem to require some nonlogical concepts = (such as=20 set membership) which cannot be defined in terms of logical concepts, at = least=20 when axiomatized by certain powerful nonlogical axioms (such as the = proper=20 axioms of ZermeloFraenkel set theory). Despite the fact that a = contradiction=20 invalidated a part of his system, the intricate theoretical web of = definitions=20 and proofs developed in the Grundgesetze nevertheless offered=20 philosophical logicians an intriguing conceptual framework. The ideas of = Bertrand Russell = and Alfred North = Whitehead=20 in Pri= ncipia=20 Mathematica owe a huge debt to the work found in Frege's=20 Grundgesetze.
Despite Frege's failure to provide a coherent systematization of the = notion=20 of an extension, we shall make use of the notion in what follows to = explain=20 Frege's theory of numbers and analysis of number statements. Though the=20 discussion will involve the notion of an extension, we shall not require = Basic=20 Law V; thus, we can use our informal understanding of the notion. In = addition,=20 extensions can be rehabilitated in various ways, either axiomatically as = in=20 modern set theory (which appears to be consistent) or as in various = consistent=20 reconstructions of Frege's system.
2.5 The Analysis of Statements of Number
In what has come to be regarded as a seminal treatise, Die = Grundlagen der=20 Arithmetik (1884), Frege began work on the idea of deriving some of = the=20 basic principles of arithmetic from what he thought were more = fundamental=20 logical principles and logical concepts. Philosophers today still find = that work=20 insightful. The leading idea is that a statement of number, such as = =91There are=20 nine planets=92 and =91There are two authors of Principia = Mathematica=92, is=20 really a statement about a concept. Frege realized that one and the same = physical phenomena could be conceptualized in different ways, and that = answers=20 to the question =91How many?=92 only make sense once a concept = F is=20 supplied. Thus, one and the same physical entity might be conceptualized = as=20 consisting of 1 army, 5 divisions, 20 regiments, 100 companies, etc., = and so the=20 question =91How many?=92 only becomes legitimate once one supplies the = concept being=20 counted, such as army, division, regiment, or = company (1884, =A746).
Using this insight, Frege took true statements like =91There are nine = planets=92=20 and =91There are two authors of Principia Mathematica=92 to be = higherorder=20 claims about the concepts planet and author of Principia=20 Mathematica, respectively. In the second case, the higherorder = claim=20 asserts that the firstorder concept being an author of Principia=20 Mathematica falls under the secondorder concept being a = concept under=20 which two objects fall. This sounds circular, since it looks like = we have=20 analyzed
There are two authors of Principia=20 Mathematica,which involves the concept two, as
The concept being an author of Principia = Mathematica=20 falls the concept being a concept under which two objects=20 fall,which also involves the concept two. But despite = appearances, there=20 is no circularity, since Frege analyzes the secondorder concept = being a=20 concept under which two objects fall without appealing to the = concept=20 two. He did this by defining =91F is a concept under = which two=20 objects fall=92, in purely logical terms, as any concept F that = satisfies=20 the following condition:
There are distinct things x and y that = fall=20 under the concept F and anything else that falls under the = concept=20 F is identical to either x or = y.In the notation of the modern predicate calculus, this is formalized = as:
∃x∃y(x≠y & Fx=20 & Fy & ∀z(Fz → = z=3Dx =20 z=3Dy))Note that the concept being an author of Principia = Mathematica=20 satisfies this condition, since there are distinct objects x = and=20 y, namely, Bertrand Russell and Alfred North Whitehead, who = authored=20 Principia Mathematica and who are such that anything else = authoring=20 Principia Mathematica is identical to one of them. In this way, = Frege=20 analyzed a statement of number (=91there are two authors of = Principia=20 Mathematica=92) as higherorder logical statements about = concepts.
Frege then took his analysis one step further. He noticed that each = of the=20 conditions in the following sequence of conditions defined a class of=20 =91equinumerous=92 concepts, where =91F=92 in each case is a = variable ranging=20 over concepts:
Condition (0): Nothing falls under F =
=AC∃xFxCondition (1): Exactly one thing falls under F =
∃x(Fx=20 & ∀y(Fy → = y=3Dx))Condition (2): Exactly two things fall under F.=20 =
∃x∃y(x≠y & = Fx &=20 Fy & ∀z(Fz → = z=3Dx =20 z=3Dy))Condition (3): Exactly three things fall under F.=20 =
∃x∃y∃z(x≠= y &=20 x≠z & y≠z = & Fx=20 & Fy & Fz & = ∀w(Fw →=20 w=3Dx =20 w=3Dy =20 w=3Dz))etc. Notice that if concepts P and Q are both concepts = which=20 satisfy one of these conditions, then there is a onetoone = correspondence=20 between the objects which fall under P and the objects which = fall under=20 Q. That is, if any of the above conditions accurately describes = both=20 P and Q, then every object falling under P = can be=20 paired with a unique and distinct object falling under Q and, = under=20 this pairing, every object falling under Q gets paired with = some unique=20 and distinct object falling under P. (By the logician's = understanding=20 of the phrase =91every=92, this last claim even applies to those = concepts P=20 and Q which satisfy Condition (0).) Frege would call such = P=20 and Q equinumerous concepts (1884, =A772). Indeed, for = each=20 condition defined above, the concepts that satisfy the condition are all = pairwise equinumerous to one another.
With this notion of equinumerosity, Frege defined =91the number of = the concept=20 F=92 (or, more informally, =91the number of Fs=92) to = be the=20 extension or set of all concepts that are equinumerous with F = (1884,=20 =A768). For example, the number of the concept author of Principia=20 Mathematica is the extension of all concepts that are equinumerous = to that=20 concept. This number is therefore identified with the class of all = concepts=20 under which two objects fall, as this is defined by Condition (2) above. = Frege=20 specifically identified the number 0 as the number of the concept = not being=20 selfidentical (1884, =A774). It is a theorem of logic that nothing = falls=20 under this concept. Thus, it is a concept that satisfies Condition (0) = above.=20 Frege thereby identified the number 0 as the class of all concepts under = which=20 nothing falls, since that is the class of concepts equinumerous with the = concept=20 not being selfidentical. Essentially, Frege identified the = number 1 as=20 the class of all concepts which satisfy Condition (1). And so forth. But = though=20 this defines a sequence of entities which are numbers, this procedure = doesn't=20 actually define the concept natural number (finite = number).=20 Frege, however, had an even deeper idea about how to do this.
2.6 Natural Numbers
In order to define the concept of natural number, Frege = first=20 defined, for every 2place relation R, the general concept = =91x=20 is an ancestor of y in the Rseries=92. This new = relation is=20 called =91the ancestral of the relation R=92. The ancestral of the = relation=20 R was first defined in Frege's Begriffsschrift (1879, = =A726,=20 Proposition 76; 1884, =A779). The intuitive idea is easily grasped if we = consider=20 the relation x is the father of y. Suppose that = a is=20 the father of b, that b is the father of c, = and that=20 c is the father of d. Then Frege's definition of = =91x=20 is an ancestor of y in the fatherhoodseries=92 ensured that = a=20 is an ancestor of b, c, and d, that = b is an=20 ancestor of c and d, and that c is an = ancestor of=20 d.
More generally, if given a series of facts of the form aRb,=20 bRc, cRd, and so on, Frege showed how to define the = relation=20 x is an ancestor of y in the Rseries (Frege referred to this = as:=20 y follows x in the Rseries). To exploit this = definition in the case of natural numbers, Frege had to define both the = relation=20 x precedes y and the ancestral of this relation, namely, x = is an=20 ancestor of y in the predecessorseries. He first defined the = relational=20 concept x precedes y as follows (1884, =A776):
x precedes y iff there is a concept F and = an=20 object z such that:=20
 z falls under F,=20
 y is the (cardinal) number of the concept F, = and=20
 x is the (cardinal) number of the concept object = other than=20 z falling under F
In the notation of the secondorder predicate calculus, augmented by = the=20 functional notation =91#F=92 to denote the number of = Fs and by the=20 λnotation =91[λu φ]=92 to name the complex = concept being a u=20 such that φ, Frege's definition becomes:
Precedes(x,y) =3D_{df<= /EM>} ∃=20 = F ∃z(Fz & y=3D#<= EM>=20 = F & x=3D#[λu Fu &= =20 u≠z])To see the intuitive idea behind this definition, consider how the = definition=20 is satisfied in the case of the number 1 preceding the number 2: there = is a=20 concept F (e.g., let F =3D being an author of = Principia=20 Mathematica) and an object z (e.g., let z =3D = Alfred North=20 Whitehead) such that:
 Whitehead falls under the concept author of Principia=20 Mathematica,=20
 2 is the (cardinal) number of the concept author of = Principia=20 Mathematica, and=20
 1 is the (cardinal) number of the concept author of = Principia=20 Mathematica other than Whitehead.
Note that the last conjunct is true because there is exactly 1 object = (namely, Bertrand Russell) which falls under the concept object = other than=20 Whitehead which falls under the concept of being an author of Principia=20 Mathematica.
Thus, Frege has a definition of precedes which applies to = the pairs=20 <0,1>, <1,2>, <2,3>,=85. Frege then defined the = ancestral of=20 this relation, namely, x is an ancestor of y in the = predecessorseries.=20 Though the exact definition will not be given here, we note that it has = the=20 following consequence: if 10 precedes 11 and 11 precedes 12, it follows = that 10=20 is an ancestor of 12 in the predecessorseries. Note, however, that = although 10=20 is an ancestor of 12, 10 does not precede 12, for the notion of=20 precedes is that of strictly precedes. Note also that = by=20 defining the ancestral of the precedence relation, Frege had in effect = defined=20 x < y.
Recall that Frege defined the number 0 as the number of the concept = not=20 being selfidentical, and that 0 thereby becomes identified with = the=20 extension of all concepts which fail to be exemplified. Using this = definition,=20 Frege defined (1884, =A783):
x is a natural number iff either x=3D0 or = 0 is an=20 ancestor of x in the predecessorseriesIn other words, a natural number is any member of the = predecessorseries=20 beginning with 0.
Using this definition as a basis, Frege later derived many important = theorems=20 of number theory. Philosophers only recently appreciated the importance = of this=20 work (C. Parsons 1965, Smiley 1981, Wright 1983, and Boolos 1987, 1990, = 1995).=20 Wright 1983 in particular showed how the Dedekind/Peano axioms for = number might=20 be derived from one of the consistent principles that Frege discussed in = 1884,=20 now known as Hume's Principle (=91The number of Fs is equal to = the number=20 of Gs if and only if there is a onetoone correspondence = between the=20 Fs and the Gs=92). It was recently shown by R. Heck = [1993] that,=20 despite the logical inconsistency in the system of Frege 1893/1903, = Frege=20 himself validly derived the the Dedekind/Peano axioms from Hume's = Principle.=20 Although Frege used his inconsistent axiom, Basic Law V, to establish = Hume's=20 Principle, once Hume's Principle was established, the subsequent = derivations of=20 the Dedekind/Peano axioms make no further essential appeals to Basic Law = V.=20 Following the lead of George Boolos, philosophers today call the = derivation of=20 the Dedekind/Peano Axioms from Hume's Principle =91Frege's Theorem=92. = The proof of=20 Frege's Theorem was a tour de force which involved some of the = most=20 beautiful, subtle, and complex logical reasoning that had ever been = devised. For=20 a comprehensive introduction to the logic of Frege's Theorem, see the = entry Frege's logic, = theorem,=20 and foundations for arithmetic.
2.7 Frege's Conception of Logic
Before receiving the famous letter from Bertrand Russell informing = him of the=20 inconsistency in his system, Frege thought that he had shown that = arithmetic is=20 reducible to the analytic truths of logic (i.e., statements which are = true=20 solely in virtue of the meanings of the logical words appearing in those = statements). It is recognized today, however, that at best Frege showed = that=20 arithmetic is reducible to secondorder logic extended only by Hume's = Principle.=20 Some philosophers think Hume's Principle is analytically true (i.e., = true in=20 virtue of the very meanings of its words), while others resist the = claim, and=20 there is an interesting debate on this issue in the literature. However, = for the=20 purposes of this introductory essay, there are prior questions on which = it is=20 more important to focus, concerning the nature of Frege's logic, namely, = =91Did=20 Frege's 1879 or 1893/1903 system (excluding Basic Law V) contain any=20 extralogical resources?=92, and =91How did Frege's conception = of logic=20 differ from that of his predecessors, and in particular, Kant's?=92 For = even if=20 Frege had been right in thinking that arithmetic is reducible to truths = of=20 logic, it is well known that Kant thought that arithmetic consisted of = synthetic=20 (a priori) truths and that it was not reducible to = analytic=20 logical truths. But, of course, Frege's view and Kant's view contradict = each=20 other only if they have the same conception of logic. Do they?
MacFarlane 2002 addresses this question, and points out that their=20 conceptions differ in various ways:
=85 the resources Frege recognizes as logical far outstrip = those of=20 Kant's logic (Aristotelian term logic with a simple theory of = disjunctive and=20 hypothetical propositions added on). The most dramatic difference is = that=20 Frege's logic allows us to define concepts using nested quantifiers, = while=20 Kant's is limited to representing inclusion relations.MacFarlane goes on to point out that Frege's logic also contains = higherorder=20 quantifiers (i.e., quantifiers ranging over concepts), and a logical = functor for=20 forming singular terms from open sentences (i.e., the expression =91the = extension=20 of=92 takes the open sentence φ(x) to yield the singular = term, =91the=20 extension of the concept φ(x)=92). MacFarlane notes that = if we were to=20 try to express such resources in Kant's system, we would have to appeal = to=20 nonlogical constructions which make sense only with respect to a = faculty of=20 =91intuition=92, that is, an extralogical source which presents our = minds with=20 (sensible) phenomena about which judgments can be formed. Frege denies = Kant's=20 dictum (A51/B75), =91Without sensibility, no object would be given to = us=92,=20 claiming that 0 and 1 are objects but that they =91can't be given to us = in=20 sensation=92 (1884, 101). (Frege's view is that our understanding can = grasp them=20 as objects if their definitions can be grounded in analytic propositions = governing extensions of concepts.)
The debate over which resources require an appeal to intuition and = which do=20 not is an important one, since Frege dedicated himself to the idea of=20 eliminating appeals to intuition in the proofs of the basic propositions = of=20 arithmetic. Frege saw himself very much in the spirit of Bolzano (1817), = who=20 eliminated the appeal to intuition in the proof of the intermediate = value=20 theorem in the calculus by proving this theorem from the definition of=20 continuity, which had recently been defined in terms of the definition = of a=20 limit (see Coffa 1991, 27). A Kantian might very well simply draw a = graph of a=20 continuous function which takes values above and below the origin, and = thereby=20 =91demonstrate=92 that such a function must cross the origin. But both = Bolzano and=20 Frege saw such appeals to intuition as potentially introducing logical = gaps into=20 proofs. There are good reasons to be suspicious about such appeals: (1) = there=20 are examples of functions which we can't graph or otherwise construct = for=20 presentation to our intuitive faculty=97consider the function =83 which = maps=20 rational numbers to 0 and irrational numbers to 1, or consider those = functions=20 which are everywhere continuous but nowhere differentiable; (2) once we = take=20 certain intuitive notions and formalize them in terms of explicit = definitions,=20 the formal definition might imply counterintuitive results; and (3) the = rules of=20 inference from statements to constructions and back are not always = clear. Frege=20 explicitly remarked upon the fact that he labored to avoid constructions = and=20 appeals to intuition in the proofs of basic propositions of arithmetic = (1879,=20 Preface/5, Part III/=A723; 1884, =A7 62, 87; 1893, =A70; and 1903, = Appendix).
This brings us to one of the most important differences between the = Frege's=20 logic and Kant's. Frege's secondorder logic included a Rule of = Substitution=20 (Grundgesetze I, 1893, =A748, item 9), which allows one to = substitute=20 complex open formulas into logical theorems to produce new logical = theorems.=20 This rule is equivalent to a very powerful existence condition governing = concepts known as the Comprehension Principle for Concepts. This = principle=20 asserts the existence of a concept corresponding to every open formula = of the=20 form φ(x) with free variable x, no matter how = complex φ is.=20 From Kant's point of view, existence claims were thought to be synthetic = and in=20 need of justification by the faculty of intuition. So, although it was = one of=20 Frege's goals to avoid appeals to the faculty of intuition, there is a = real=20 question as to whether his system, which involves an inference rule = equivalent=20 to a principle asserting the existence of a wide range of concepts, = really is=20 limited in its scope to purely logical laws of an analytic nature.
One final important difference between Frege's conception of logic = and Kant's=20 concerns the question of whether logic has any content unique to itself. = As=20 MacFarlane 2002 points out, one of Kant's most central views about logic = is that=20 its axioms and theorems are purely formal in nature, i.e., abstracted = from all=20 semantic content and concerned only with the forms of judgments, which = are=20 applicable across all the physical and mathematical sciences (1781/1787, = A55/B79, A56/B80, A70/B95). By contrast, Frege took logic to have its = own unique=20 subject matter, which included not only facts about concepts (concerning = negation, subsumption, etc.), identity, etc. (Frege 1906, 428), but also = facts=20 about ancestrals of relations and natural numbers (1879, 1893). Logic is = not=20 purely formal, from Frege's point of view, but rather can provide = substantive=20 knowledge of objects and concepts.
Despite these fundamental differences in their conceptions of logic, = Kant and=20 Frege may have agreed that the most important defining characteristic of = logic=20 is its generality, i.e., the fact that it provides norms (rules, = prescriptions)=20 that are constitutive of thought. This rapprochement between Kant and = Frege is=20 developed in some detail in MacFarlane 2002. The reader will find there = reasons=20 for thinking that Kant and Frege may have shared enough of a common = conception=20 about logic for us to believe that equivocation doesn't undermine the = apparent=20 inconsistency between their views on the reducibility of arithmetic to = logic. It=20 is by no means settled as to how we should think of the relationship = between=20 arithmetic and logic, since logicians have not yet come to agreement = about the=20 proper conception of logic. Many modern logicians have a conception of = logic=20 that is yet different from both Kant and Frege. It is one which evolves = out of=20 the ideas that (1) certain concepts and laws remain invariant under = permutations=20 of the domain of quantification, and (2) that logic ought not to dictate = the=20 size of the domain of quantification. But this conception has not yet = been=20 articulated in a widely accepted way, and so elements common to Frege's = and=20 Kant's conception may yet play a role in our understanding of what logic = is.=20 (For an excellent discussion of Frege's conception of logic, see = Goldfarb=20 2001.)
3. Frege's Philosophy of Language
While pursuing his investigations into mathematics and logic (and = quite=20 possibly, in order to ground those investigations), Frege was led to = develop a=20 philosophy of language. His philosophy of language has had just as much, = if not=20 more, impact than his contributions to logic and mathematics. Frege's = seminal=20 paper in this field =91=DCber Sinn und Bedeutung=92 (=91On Sense and = Reference=92, 1892a)=20 is now a classic. In this paper, Frege considered two puzzles about = language and=20 noticed, in each case, that one cannot account for the meaningfulness or = logical=20 behavior of certain sentences simply on the basis of the denotations of = the=20 terms (names and descriptions) in the sentence. One puzzle concerned = identity=20 statements and the other concerned sentences with relative clauses such = as=20 propositional attitude reports. To solve these puzzles, Frege suggested = that the=20 terms of a language have both a sense and a denotation, i.e., that at = least two=20 semantic relations are required to explain the significance or meaning = of the=20 terms of a language. This idea has inspired research in the field for = over a=20 century.
3.1 Frege's Puzzles
3.1.1 Frege's Puzzle About Identity Statements
Here are some examples of identity statements:
117+136 =3D 253.
The morning star is identical to the = evening=20 star.
Mark Twain is Samuel Clemens.
Bill is Debbie's=20 father.Frege believed that these statements all have the form = =91a=3Db=92, where=20 =91a=92 and =91b=92 are either names or descriptions = that=20 denote individuals. He naturally assumed that a sentence of the = form=20 =91a=3Db=92 is true if and only the object a = just is=20 (identical to) the object b. For example, the sentence = =91117+136 =3D 253=92=20 is true if and only if the number 117+136 just is the number 253. And = the=20 statement =91Mark Twain is Samuel Clemens=92 is true if and only if the = person Mark=20 Twain just is the person Samuel Clemens.
But Frege noticed (1892) that this account of truth can't be all = there is to=20 the meaning of identity statements. The statement =91a=3Da=92 = has a cognitive=20 significance (or meaning) that must be different from the cognitive = significance=20 of =91a=3Db=92. We can learn that =91Mark Twain=3DMark Twain=92 = is true simply by=20 inspecting it; but we can't learn the truth of =91Mark Twain=3DSamuel = Clemens=92=20 simply by inspecting it  you have to examine the world to see whether = the two=20 persons are the same. Similarly, whereas you can learn that =91117+136 = =3D 117+136=92=20 and =91the morning star is identical to the morning star=92 are true = simply by=20 inspection, you can't learn the truth of =91117+136 =3D 253=92 and = =91the morning star=20 is identical to the evening star=92 simply by inspection. In the latter = cases, you=20 have to do some arithmetical work or astronomical investigation to learn = the=20 truth of these identity claims. Now the problem becomes clear: the = meaning of=20 =91a=3Da=92 clearly differs from the meaning of = =91a=3Db=92, but given the=20 account of the truth described in the previous paragraph, these two = identity=20 statements appear to have the same meaning whenever they are true! For = example,=20 =91Mark Twain=3DMark Twain=92 is true just in case: the person Mark = Twain is identical=20 with the person Mark Twain. And =91Mark Twain=3DSamuel Clemens=92 is = true just in=20 case: the person Mark Twain is identical with the person Samuel Clemens. = But=20 given that Mark Twain just is Samuel Clemens, these two cases are the = same case,=20 and that doesn't explain the difference in meaning between the two = identity=20 sentences. And something similar applies to all the other examples of = identity=20 statements having the forms =91a=3Da=92 and = =91a=3Db=92.
So the puzzle Frege discovered is: how do we account for the = difference in=20 cognitive significance between =91a=3Db=92 and = =91a=3Da=92 when they are=20 true?
3.1.2 Frege's Puzzle About Propositional Attitude Reports
Frege is generally credited with identifying the following puzzle = about=20 propositional attitude reports, even though he didn't quite describe the = puzzle=20 in the terms used below. A propositional attitude is a psychological = relation=20 between a person and a proposition. Belief, desire, intention, = discovery,=20 knowledge, etc., are all psychological relationships between persons, on = the one=20 hand, and propositions, on the other. When we report the propositional = attitudes=20 of others, these reports all have a similar logical form:
x believes that p
x desires = that=20 p
x intends that p
x = discovered=20 that p
x knows that pIf we replace the variable =91x =92 by the name of a person = and replace=20 the variable =91p =92 with a sentence that describes the = propositional=20 object of their attitude, we get specific attitude reports. So by = replacing=20 =91x =92 by =91John=92 and =91p =92 by =91Mark Twain = wrote Huckleberry=20 Finn=92 in the first example, the result would be the following = specific=20 belief report:
John believes that Mark Twain wrote Huckleberry=20 Finn.To see the problem posed by the analysis of propositional attitude = reports,=20 consider what appears to be a simple principle of reasoning, namely, the = Principle of Substitution. If a name, say n, appears in a true = sentence=20 S, and the identity sentence n=3Dm is true, then the Principle = of=20 Substitution tells us that the substitution of the name m for = the name=20 n in S does not affect the truth of S. For example, let S be = the true=20 sentence =91Mark Twain was an author=92, let n be the name = =91Mark Twain=92,=20 and let m be the name =91Samuel Clemens=92. Then since the = identity=20 sentence =91Mark Twain=3DSamuel Clemens=92 is true, we can substitute = =91Samuel Clemens=92=20 for =91Mark Twain=92 without affecting the truth of the sentence. And = indeed, the=20 resulting sentence =91Samuel Clemens was an author=92 is true. In other = words, the=20 following argument is valid:
Mark Twain was an author.
Mark Twain=3DSamuel Clemens.=20
Therefore, Samuel Clemens was an author.Similarly, the following argument is valid.
4 > 3
4=3D8/2
Therefore, 8/2 > 3In general, then, the Principle of Substitution seems to take the = following=20 form, where S is a sentence, n and m are names, and=20 S(n) differs from S(m) only by the fact that at least = one=20 occurrence of m replaces n:
From S(n) and n=3Dm, infer=20 S(m)This principle seems to capture the idea that if we say something = true about=20 an object, then even if we change the name by which we refer to that = object, we=20 should still be saying something true about that object.
But Frege, in effect, noticed the following counterexample to the = Principle=20 of Substitution. Consider the following argument:
John believes that Mark Twain wrote Huckleberry = Finn.=20
Mark Twain=3DSamuel Clemens.
Therefore, John believes that = Samuel=20 Clemens wrote Huckleberry Finn.This argument is not valid. There are circumstances in which the = premises are=20 true and the conclusion false. We have already described such = circumstances,=20 namely, one in which John learns the name =91Mark Twain=92 by reading=20 Huckleberry Finn but learns the name =91Samuel Clemens=92 in = the context of=20 learning about 19th century American authors (without learning that the = name=20 =91Mark Twain=92 was a pseudonym for Samuel Clemens). John may = not believe=20 that Samuel Clemens wrote Huckleberry Finn. The premises of the = above=20 argument, therefore, do not logically entail the conclusion. So the = Principle of=20 Substitution appears to break down in the context of propositional = attitude=20 reports. The puzzle, then, is to say what causes the principle to fail = in these=20 contexts. Why aren't we still saying something true about the man in = question if=20 all we have done is changed the name by which we refer to him?
3.2 Frege's Theory of Sense and Denotation
To explain these puzzles, Frege suggested that in addition to having = a=20 denotation, names and descriptions also express a sense. The = sense of=20 an expression accounts for its cognitive significance=97it is the way by = which one=20 conceives of the denotation of the term. The expressions =914=92 and = =918/2=92 have the=20 same denotation but express different senses, different ways of = conceiving the=20 same number. The descriptions =91the morning star=92 and =91the evening = star=92 denote=20 the same planet, namely Venus, but express different ways of conceiving = of Venus=20 and so have different senses. The name =91Pegasus=92 and the description = =91the most=20 powerful Greek god=92 both have a sense (and their senses are distinct), = but=20 neither has a denotation. However, even though the names =91Mark = Twain=92 and=20 =91Samuel Clemens=92 denote the same individual, they express different = senses.=20 Using the distinction between sense and denotation, Frege can account = for the=20 difference in cognitive significance between identity statements of the = form=20 =91a=3Da=92 and =91a=3Db=92. Since the sense of = =91a=92 differs from=20 the sense of =91b=92, the components of the sense of = =91a=3Da=92 and the=20 sense of =91a=3Db=92 are different, guaranteeing that the sense = of the whole=20 expression will be different in the two cases. Since the sense of an = expression=20 accounts for its cognitive significance, Frege has an explanation of the = difference in cognitive significance between =91a=3Da=92 and = =91a=3Db=92,=20 and thus a solution to the first puzzle.
Moreover, Frege proposed that when a term (name or description) = follows a=20 propositional attitude verb, it no longer denotes what it ordinarily = denotes.=20 Instead, Frege claims that in such contexts, a term denotes its ordinary = sense.=20 This explains why the Principle of Substitution fails for terms = following the=20 propositional attitude verbs in propositional attitude reports. The = Principle=20 asserts that truth is preserved when we substitute one name for another = having=20 the same denotation. But, according to Frege's theory, the names =91Mark = Twain=92=20 and =91Samuel Clemens=92 denote different senses when they occur in the = following=20 sentences:
John believes that Mark Twain wrote Huckleberry = Finn.=20
John believes that Samuel Clemens wrote Huckleberry=20 Finn.If they don't denote the same object, then there is no reason to = think that=20 substitution of one name for another would preserve truth.
Frege developed the theory of sense and denotation into a = thoroughgoing=20 philosophy of language. This philosophy can be explained, at least in = outline,=20 by considering a simple sentence such as =91John loves Mary=92. In = Frege's view, the=20 words =91John=92 and =91Mary=92 in this sentence are names, the = expression =91loves=92=20 signifies a function, and, moreover, the sentence as a whole is a = complex name.=20 Each of these expressions has both a sense and a denotation. The sense = and=20 denotation of the names are basic; but sense and denotation of the = sentence as a=20 whole can be described in terms of the sense and denotation of the names = and the=20 way in which those words are arranged in the sentence alongside the = expression=20 =91loves=92. Let us refer to the denotation and sense of the words as = follows:
d[j] refers to the denotation of the name = =91John=92.=20
d[m] refers to the denotation of the name = =91Mary=92.=20
d[L] refers to the denotation of the expression = =91loves=92.=20
s[j] refers to the sense of the name =91John=92.=20
s[m] refers to the sense of the name =91Mary=92.=20
s[L] refers to the sense of the expression=20 =91loves=92.We now work toward a theoretical description of the denotation of the = sentence as a whole. On Frege's view, d[j] and=20 d[m] are the real individuals John and Mary, = respectively.=20 d[L] is a function that maps d[m] = (i.e., Mary)=20 to a function which serves as the denotation of the predicate =91loves = Mary=92. Let=20 us refer to that function as d[Lm]. Now the function=20 d[Lm] maps d[j] (i.e., John) to the = denotation=20 of the sentence =91John loves Mary=92. Let us refer to the denotation of = the=20 sentence as d[jLm]. Frege identifies the denotation of = a=20 sentence as one of the two truth values. Because d[Lm] = maps=20 objects to truth values, it is a concept. Thus, d[jLm] = is the=20 truth value The True if John falls under the concept = d[Lm];=20 otherwise it is the truth value The False. So, on Frege's view, the = sentence=20 =91John loves Mary=92 names a truth value.
The sentence =91John loves Mary=92 also expresses a sense. Its sense = may be=20 described as follows. Although Frege doesn't appear to have explicitly = said so,=20 his work suggests that s[L] (the sense of the = expression=20 =91loves=92) is a function. This function would map = s[m] (the sense=20 of the name =91Mary=92) to the sense of the predicate =91loves Mary=92. = Let us refer to=20 the sense of =91loves Mary=92 as s[Lm]. Now again, = Frege's work=20 seems to imply that we should regard s[Lm] as a = function which=20 maps s[j] (the sense of the name =91John=92) to the = sense of the=20 whole sentence. Let us call the sense of the entire sentence=20 s[jLm]. Frege calls the sense of a sentence a = thought,=20 and whereas there are only two truth values, he supposes that there are = an=20 infinite number of thoughts.
With this description of language, Frege can give a general account = of the=20 difference in the cognitive significance between identity statements of = the form=20 =91a=3Da=92 and =91a=3Db=92. The = cognitive significance=20 is not accounted for at the level of denotation. On Frege's view, the = sentences=20 =914=3D8/2=92 and =914=3D4=92 both denote the same truth value. The = function=20 ( )=3D( ) maps 4 and 8/2 to The True, i.e., maps 4 and 4 to = The True. So=20 d[4=3D8/2] is identical to d[4=3D4]; = they are both=20 The True. However, the two sentences in question express different = thoughts.=20 That is because s[4] is different from = s[8/2].=20 So the thought s[4=3D8/2] is distinct from the thought=20 s[4=3D4]. Similarly, =91Mark Twain=3DMark Twain=92 and = =91Mark=20 Twain=3DSamuel Clemens=92 denote the same truth value. However, given = that=20 s[Mark Twain] is distinct from = s[Samuel=20 Clemens], Frege would claim that the thought s[Mark = Twain=3DMark=20 Twain] is distinct from the thought s[Mark = Twain=3DSamuel=20 Clemens].
Furthermore, recall that Frege proposed that terms following = propositional=20 attitude verbs denote not their ordinary denotations but rather the = senses they=20 ordinarily express. In fact, in the following propositional attitude = report, not=20 only do the words =91Mark Twain=92, =91wrote=92 and =91Huckleberry = Finn =92 denote=20 their ordinary senses, but the entire sentence =91Mark Twain wrote = Huckleberry=20 Finn=92 also denotes its ordinary sense (namely, a thought):
John believes that Mark Twain wrote Huckleberry=20 Finn.Frege, therefore, would analyze this attitude report as follows: = =91believes=20 that=92 denotes a function that maps the denotation of the sentence = =91Mark Twain=20 wrote Huckleberry Finn=92 to a concept. In this case, however, = the=20 denotation of the sentence =91Mark Twain wrote Huckleberry = Finn=92 is not a=20 truth value but rather a thought. The thought it denotes is different = from the=20 thought denoted by =91Samuel Clemens wrote Huckleberry Finn=92 = in the=20 following propositional attitude report:
John believes that Samuel Clemens wrote Huckleberry=20 Finn.Since the thought denoted by =91Samuel Clemens wrote Huckleberry = Finn=92=20 in this context differs from the thought denoted by =91Mark Twain wrote=20 Huckleberry Finn=92 in the same context, the concept denoted by = =91believes=20 that Mark Twain wrote Huckleberry Finn=92 is a different = concept from the=20 one denoted by =91believes that Samuel Clemens wrote Huckleberry = Finn=92.=20 One may consistently suppose that the concept denoted by the former = predicate=20 maps John to The True whereas the the concept denoted by the latter = predicate=20 does not. Frege's analysis therefore preserves our intuition that John = can=20 believe that Mark Twain wrote Huckleberry Finn without = believing that=20 Samuel Clemens did. It also preserves the Principle of = Substitutionthe fact=20 that one cannot substitute =91Samuel Clemens=92 for =91Mark Twain=92 = when these names=20 occur after propositional attitude verbs does not constitute evidence = against=20 the Principle. For if Frege is right, names do not have their usual = denotation=20 when they occur in these contexts.
Bibliography
A. Primary Sources
Chronological Catalog of Frege's Work (PDF)
Works by Frege Cited in this Entry
1879 Begriffsschrift, eine der arithmetischen nachgebildete=20 Formelsprache des reinen Denkens, Halle a. S.: Louis Nebert.=20 Translated as Concept Script, a formal language of pure = thought=20 modelled upon that of arithmetic, by S. BauerMengelberg in = J.=20 vanHeijenoort (ed.), From Frege to G=F6del: A Source Book in=20 Mathematical Logic, 18791931, Cambridge, MA: Harvard = University=20 Press, 1967. 1884 Die Grundlagen der Arithmetik: eine logischmathematische=20 Untersuchung =FCber den Begriff der Zahl, Breslau: W. = Koebner.=20 Translated as The Foundations of Arithmetic: A = logicomathematical=20 enquiry into the concept of number, by J.L. Austin, Oxford:=20 Blackwell, second revised edition, 1974. 1891 =91Funktion und Begriff=92, Vortrag, gehalten in der Sitzung vom = 9. Januar=20 1891 der Jenaischen Gesellschaft f=FCr Medizin und = Naturwissenschaft, Jena:=20 Hermann Pohle. Translated as =91Function and Concept=92 by P. = Geach in=20 Translations from the Philosophical Writings of Gottlob = Frege, P.=20 Geach and M. Black (eds. and trans.), Oxford: Blackwell, third = edition,=20 1980. 1892a =91=DCber Sinn und Bedeutung=92, in Zeitschrift f=FCr = Philosophie und=20 philosophische Kritik, 100: 2550. = Translated as =91On=20 Sense and Reference=92 by M. Black in Translations from the=20 Philosophical Writings of Gottlob Frege, P. Geach and M. = Black (eds.=20 and trans.), Oxford: Blackwell, third edition, 1980. 1892b =91=DCber Begriff und Gegenstand=92, in Vierteljahresschrift = f=FCr=20 wissenschaftliche Philosophie, 16: 192205.=20 Translated as =91Concept and Object=92 by P. Geach in = Translations from=20 the Philosophical Writings of Gottlob Frege, P. Geach and M. = Black=20 (eds. and trans.), Oxford: Blackwell, third edition, = 1980. 1893 Grundgesetze der Arithmetik, Jena: Verlag Hermann = Pohle, Band=20 I. Partial translation as The Basic Laws of Arithmetic by = M.=20 Furth, Berkeley: U. of California Press, 1964. 1903 Grundgesetze der Arithmetik, Jena: Verlag Hermann = Pohle, Band=20 II. 1904 =91Was ist eine Funktion?=92, in Festschrift Ludwig = Boltzmann gewidmet=20 zum sechzigsten Geburtstage, 20. Februar 1904, S. Meyer = (ed.),=20 Leipzig: Barth, 1904, pp. 656666. Translated as =91What is a = Function?=92 by=20 P. Geach in Translations from the Philosophical Writings of = Gottlob=20 Frege, P. Geach and M. Black (eds. and trans.), Oxford: = Blackwell,=20 third edition, 1980. 1906 =91=DCber die Grundlagen der Geometrie=92 (Second Series), = Jahresbericht=20 der Deutschen MathematikerVereinigung 15, = pp.=20 293309 (Part I), 377403 (Part II), 423430 (Part III). = Translation as=20 =91On the Foundations of Geometry (Second Series)=92 by E.H. W. = Kluge, in=20 On the Foundatons of Geometry and Formal Theories of = Arthmetic,=20 New Haven: Yale University Press, 1971. B. Secondary Sources
 Beaney, M., 1996, Frege: Making Sense, London: Duckworth. =
 Bell, D., 1979, Frege's Theory of Judgment, Oxford: = Clarendon.=20
 Bolzano, B., 1817, =91Rein analytischer Beweis des Lehrsatzes=92, = in Early=20 Mathematical Works (1781=961848), L. Novy (ed.), Institute of = Czechoslovak=20 and General History CSAS, Prague, 1981.=20
 Boolos, G., 1986, =91Saving Frege From Contradiction=92, = Proceedings of=20 the Aristotelian Society, 87 (1986/87): 137151. =
 Boolos, G., 1987, =91The Consistency of Frege's Foundations of = Arithmetic=92, in J. Thomson (ed.), On Being and Saying, = Cambridge, MA: The MIT Press, pp. 320.=20
 Boolos, G., 1990, =91The Standard of Equality of Numbers=92, in G. = Boolos=20 (ed.), Meaning and Method: Essays in Honor of Hilary Putnam,=20 Cambridge: Cambridge University Press, 26177.=20
 Boolos, G., 1995, =91Frege's Theorem and the Peano Postulates=92, = The=20 Bulletin of Symbolic Logic 1, 31726.=20
 Boolos, G., 1998, Logic, Logic, and Logic, Cambridge, MA: = Harvard=20 University Press.=20
 Coffa, J.A., 1991, The Semantic Tradition from Kant to = Carnap, L.=20 Wessels (ed.), Cambridge: Cambridge University Press.=20
 Currie, G., 1982, Frege: An Introduction to His = Philosophy,=20 Brighton, Sussex: Harvester Press.=20
 Demopoulos, W., (ed.), 1995, Frege's Philosophy of = Mathematics,=20 Cambridge, MA: Harvard.=20
 Dummett, M., 1973, Frege: Philosophy of Language, London: = Duckworth.=20
 Dummett, M., 1981, The Interpretation of Frege's = Philosophy,=20 Cambridge, MA: Harvard University Press.=20
 Dummett, M., 1991, Frege: Philosophy of Mathematics, = Cambridge,=20 MA: Harvard University Press.=20
 Goldfarb, W., 2001, =91Frege's Conception of Logic=92, in J. Floyd = and S.=20 Shieh (eds.), Future Pasts: The Analytic Tradition in = TwentiethCentury=20 Philosophy, Oxford: Oxford University Press, 2541.=20
 Haaparanta, L., and Hintikka, J., (eds.), 1986, Frege=20 Synthesized, Dordrecht: D. Reidel.=20
 Heck, R., 1993, =91The Development of Arithmetic in Frege's = Grundgesetze=20 der Arithmetik=92, Journal of Symbolic Logic,=20 58/2 (June): 579601.=20
 Hodges, W., 2001, =91Formal Features of Compositionality=92, = Journal of=20 Logic, Language and Information, 10: 728.=20
 Kant, I., 1781, Kritik der reinen Vernunft, Riga: Johann=20 Friedrich Hartknoch, 1st edition (A), 1781; 2nd edition (B), 1787. = Translated=20 as Critique of Pure Reason by P. Guyer and A. Wood, = Cambridge:=20 Cambridge University Press, 1998.=20
 Klemke, E. D. (ed.), 1968, Essays on Frege, Urbana, IL:=20 University of Illinois Press.=20
 MacFarlane, J., 2002, =91Frege, Kant, and the Logic in = Logicism=92,=20 Philosophical Review, 111/1 (January): 2566.=20
 Parsons, C., 1965, =91Frege's Theory of Number=92, in M. Black = (ed.),=20 Philosophy in America, Ithaca: Cornell, 180203.=20
 Parsons, T., 1981, =91Frege's Hierarchies of Indirect Senses and = the Paradox=20 of Analysis=92, Midwest Studies in Philosophy: VI, = Minneapolis:=20 University of Minnesota Press, pp. 3757.=20
 Parsons, T., 1987, =91On the Consistency of the FirstOrder = Portion of=20 Frege's Logical System=92, Notre Dame Journal of Formal = Logic,=20 28/1 (January): 161168.=20
 Parsons, T., 1982, =91Fregean Theories of Fictional Objects=92,=20 Topoi, 1: 8187.=20
 Pelletier, F.J., 2001, =91Did Frege Believe Frege's Principle=92, = Journal=20 of Logic, Language, and Information, 10/1: = 87114.=20
 Perry, J., 1977, =91Frege on Demonstratives=92, Philosophical = Review,=20 86 (1977): 474497.=20
 Resnik, M., 1980, Frege and the Philosophy of = Mathematics,=20 Ithaca, NY: Cornell University Press.=20
 Ricketts, T., 1997, =91TruthValues and CoursesofValue in = Frege's=20 Grundgesetze=92, in Early Analytic Philosophy, W. = Tait (ed.),=20 Chicago: Open Court, pp. 187211.=20
 Ricketts, T., 1986, =91Logic and Truth in Frege=92, = Proceedings of the=20 Aristotelian Society, Supplementary Volume 70, pp. 121140.=20
 Ricketts, T., forthcoming, Cambridge Companion to Frege,=20 Cambridge: Cambridge University Press.=20
 Salmon, N., 1986, Frege's Puzzle, Cambridge, MA: MIT = Press.=20
 Schirn, M., (ed.), 1996, Frege: Importance and Legacy, = Berlin: de=20 Gruyter.=20
 Sluga, H., 1980, Gottlob Frege, London: Routledge and = Kegan Paul.=20
 Sluga, H. (ed.), 1993, The Philosophy of Frege, New York: = Garland, four volumes.=20
 Smiley, T., 1981, =91Frege and Russell=92, Epistemologica = 4: 538.=20
 Wright, C., 1983, Frege's Conception of Numbers as = Objects,=20 Aberdeen: Aberdeen University Press.
Other Internet Resources
 Die Grundlagen der Arithmetik, (528 KB PDF file), = original=20 German text (maintained by Alain Blachair, Acad=E9mie de NancyMetz)=20
 MacTutor History of Mathematics Archive=20
 Metaphysics=20 Research Lab Web Page on Frege=20
 Gottlob=20 Frege, Jena, und die Geburt der modernen Logik (Werner Stelzner, = Jena)=20
 Frege,=20 Gottlob, by Kevin Klement (U. Massachusetts/Amherst), in the = Internet=20 Encyclopedia of Philosophy.
Related Entries
Frege, Gottlob: = logic,=20 theorem, and foundations for arithmetic  logic: = classical =20 logic: intensional  logicism  mathematics, philosophy of  neologicism =  Pri= ncipia=20 Mathematica  quantification  reference  Russell, = Bertrand  Russell's = paradox=20  sense/reference distinction=20Acknowledgements
I would like to thank Kai Wehmeier, whose = careful eye=20 as a logician and Frege scholar caught several passages where I had bent = the=20 truth past the breaking point.
Edward N. Zalta
zalta@stanford.edu
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Stanford Encyclopedia of Philosophy
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