In this section, we describe the language and logic of Frege's predicate calculus. We explain his function-argument 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 second-order logic, and we introduce and explain λ-notation 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.

1.1 The Language

In Begr, Frege invented the predicate calculus. It will soon become clear that the language and logic of his predicate calculus are ‘second-order’. 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 truth-values (The True and The False), he defined a concept to be any function that always maps its arguments to truth-values. For example, whereas ‘x² +3’ and ‘father-of(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 truth-values.

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 greater-than 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 courses-of-values, and certain other letters as placeholders in the names of functions, then Frege's notation for the logical notions ‘not’, ‘if-then’, ‘every’ and ‘some’ can be described in the following table:

Logical Notion	Modern Notation	Frege-Style 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 ‘2² = 4’ names The True, ‘2² = 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 ‘second-order’.

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Thanks for this text to : Stanford Encyclopedia of Philosophy

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