
In this section, we
describe the language and logic of Frege's predicate calculus. We
explain his functionargument analysis of atomic sentences and his
definition of concepts in terms of functions, give examples of his
‘concept script’, and discuss the Rule of Substitution in his logic.
We also show how Frege's Rule of Substitution corresponds to a
comprehension principle for concepts in secondorder logic, and we
introduce and explain λnotation to help us distinguish open formulas
and complex names of concepts. Readers who are already familiar with
these ideas may wish to skip ahead to Section 2.
In Begr,
Frege invented the predicate calculus. It will soon become clear that
the language and logic of his predicate calculus are ‘secondorder’.
The language included not only the variables x,y,z, … ,
which range over objects, but also included the variables ƒ,g,h, … ,
which range over functions. Frege rigidly distinguished
objects from functions and so we may think of these variables as
ranging over separate, mutually exclusive domains. Frege took
functional application ‘ƒ(x)’ as the principal operation for
forming complex names of objects in his language. The expression ‘ƒ(x)’
denotes the object to which the function ƒ maps the object x.
Frege called the object x the ‘argument’ of the function ƒ
and called ƒ(x) the ‘value’ of the function. Since Frege also
recognized two special objects he called truthvalues (The
True and The False), he defined a concept to be any function
that always maps its arguments to truthvalues. For example, whereas ‘x^{2}
+3’ and ‘fatherof(x)’ denote ordinary functions, the
expressions ‘Happy(x)’ and ‘x > 5’ denote concepts.
The former denotes a concept which maps any object that is happy to
The True and all other objects to The False; the latter denotes a
concept that maps any object that is greater than 5 to The True and
all other objects to The False. Given that concepts like being
happy and being greater than 5 map their arguments to
truth values, the atomic sentences of Frege's language, such as
‘Happy(b)’ and ‘4 > 5’, become names of
truthvalues.
In what follows, we use
the symbols F,G, … as variables ranging over
concepts and we often write ‘Fx’ (instead of ‘F(x)’)
to express the claim that concept F maps x to The
True. When this claim is true, Frege would say that x
falls under the concept F.
When ƒ is a function of
two arguments x and y and ƒ always maps its pair of
arguments to a truth value, Frege would say that ƒ is a relation. We
shall use the expression ‘Rxy’ (or sometimes ‘R(x,y)’)
to assert that the relation R maps x and y
(in that order) to The True. In what follows, we shall sometimes write
the symbol that denotes a mathematical relation in the usual ‘infix’
notation; for example, ‘>’ denotes the greaterthan relation in the
expression ‘x > y’.
Now that we have
explained Frege's analysis of the atomic statements ‘Fx’ and
‘Rxy’ familiar to modern students of logic, we turn next to
the more complex statements of his language. Frege developed his own
graphical notation for asserting complex statements involving
negations, conditionals, and universal quantification. If we ignore
the fact that Frege used Gothic letters as variables of
quantification, certain letters as bound variables in names of
coursesofvalues, and certain other letters as placeholders in the
names of functions, then Frege's notation for the logical notions
‘not’, ‘ifthen’, ‘every’ and ‘some’ can be described in the following
table:
Logical Notion 
Modern Notation 
FregeStyle Notation 
It is not the case that Fx 
¬Fx 

If Fx then Gy 
Fx → Gy 

Every x is such
that Fx 
∀xFx 

Some x is such that
Fx 
¬∀x¬Fx,
i.e., ∃xFx 

Every F is such
that Fa 
∀F Fa 

Some F is such that
Fa 
¬∀F¬Fa,
i.e., ∃F Fa 

So, for example, whereas a modern logician
would symbolize the claim ‘All As are Bs’ as:
∀x(Ax →
Bx)
Frege would symbolize this claim as
follows:
However, since Frege's
notation was never adopted as a standard, we shall instead use the
more familiar modern notation in the remainder of this essay. [See
Beaney (1997, Appendix 2), Furth (1967), and Reck & Awodey (2004,
26–34) for a more detailed introduction to Frege's notation.] We shall
assume that the reader is familiar with the fact that negations (‘¬φ’)
and conditionals (‘φ → ψ’) can be used to define the other molecular
formulas such as conjunctions (‘φ & ψ’), disjunctions (‘φ v ψ’), and
biconditionals (‘φ ≡ ψ’). Moreover, it is important to mention that
Frege took identity statements of the form ‘x = y’
as primitive in his language. Whereas ‘2^{2} = 4’ names The
True, ‘2^{2} = 3’ names The False. The statement form ‘ƒ(x)
= y’ plays an important role in Frege's axioms and
definitions. Note finally that since Frege allowed quantification over
both objects and functions, the language of his predicate calculus
becomes ‘secondorder’.
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