Chapter 9 Survey of Symbolic Logic

CHAPTER 9

SURVEY OF SYMBOLIC LOGIC 72.

IN'fRODUCTION

In this survey of symbolic logic, we shall submit to rather severe restrictions. In the first place, our attention will be concentrated on (classical, or two-valued) elementary logic, as this system has proved in many cases an adequate tool for research in formal axiomatics; but in the last sections of this chapter, a few words will be said on the logic of higher order and on the search for a logica magna. Secondly, a detailed treatment will be given only of a restricted version of elementary logic, the so-called theory of quantification; for certain more extended versions of elementary logic, we are content with a rather summary description; except for a few additional complications, they can be treated very much on the same lines as quantification theory. For more detail, the reader may refer to recent publications by Hilbert-Ackermann (revised edition, 1958), P. C. Rosenbloom (1950), W. V. Quine (1951), Hermes-Scholz (1952), S. C. Kleene (Hl5:2), J. B. Rosser (1953), and A. Church (1956). 73.

SENTENTIAL LOGIC

In constructing sentential (or propositional) logic (which is also denoted as the sentential or propositional calculus), we use a number of sentential connectives. namely, - , v, &, -?, and ~ [sometimes also j], and an infinite sequence of atoms (atomic formulas or sentential variables) PI- P2, ... , Pk' ... (or p, g, r, ... ). Starting from the atoms, we first construct expressions (or formulas) U, V, w, ..., in accordance with the following rules: (E 1) (E 2) (E 3) U ~ V (E 4) account

Every atom Pk is an expression; If U is an expression, then [J is an expression; If U and V are expressions, then U v V, U & V, U --+ V, [and, eventually. also UfV] are expressions; Nothing is an expression (of sentential logic), except on of rules (E 1-3). 20:2

203

SENTENTIAL LOGIC

By virtue of these rules, we can draw for every expression a pedigree as described in Section 60, which shows the manner in which it has been constructed. In order to exhibit the order in which the different sentential connectives have been applied, we must introduce a system of punctuation by means of dots and brackets; however, we shall not explicitly state the rules of punctuation, as it will be shown in Section 95 how the need for a system of punctuation can be avoided. The existence of a pedigree provides a justification for the following definition of the notion of substitution. (S 1) The result W/Pi'Pk of substituting an expression W for the atom Pi in the atom Pk is (i) W, if j=k; (ij) PIC' if j=l=k; (S 2) The result W /Pi' U of substituting an expression W for the atom Pi in the expression U is U*, U* being the result W/Pi' U of substituting the expression W for the atom Pi in the expression U; (S3) The result W/Pi'(UV V) of substituting the expression W for the atom Pi in the expression U v V is U* v V*, U* and V* being respectively the substitution results W/Pi' U and W/Pi' V; similarly for the sentential connectives &, --'>-, and **; (S 4) Nothing is a substitution result exo, pt on account of the rules (S 1-3). It will be clear that, for every expression W, for every atom Pi' and for every expression U, there is a unique substitution result W/Pi' U, which, moreover, is always an expression. From now on we shall, whenever this is possible, simplify our notation by writing P, q, r, ... instead of PI> P2' P3' .... We can in the following manner introduce the notion of a thesis (of sentential logic). (T 1)

The expressions:

(I) (II)

P--'>- (q --'>- p); [p--'>-(p--'>-q)]--'>-(p--'>-q);

(III)

(p

(IV)

(p <-+ q) -'>- (p -'>- q) ;

--'>- q) ~

[(q

--'>- r) --'>-

(p ** q)

-'>-

(q --'>- p);

(VI)

(p

-'>-

q)

-'>-

[(q

(VII)

(ij

-'>-

p) -'>- (p -'>- q);

(V)

-'>-

p)

-'>-

(p

--'>- r)];

(P'-- q)];

204

SURVEY OF SYMBOLIC LOGre

(VIII) (IX) (X)

are theses;

** (p --+ q) ; (p & q) ** (p v ij); (pJq) ** (p v ij) ; (p v q)

(T 2) Let U be any thesis, let p be any atom, and let W be any expression; then W ls- U will be a thesis; (T 3) If both U and U --+ V are theses, then V is a thesis; (T 4) Nothing is a thesis except on account of rules (T 1-3). The above axioms (I)-(X) for sentential logic have been given by Tarski. Example 1. Point out the existence of a pedigree for every expression U of sentential logic. Example 2.

Justify the above definition of the notion of substitution.

Example 3. Show that, for every thesis U of sentential logic, we can draw a pedigree which shows why, on account of the rules (T 1-3), it is a thesis [such a pedigree is called a formal derivation of the thesis U from the axioms (I-X)]. Can we expect this pedigree to be unique? Example 4. Justify the following method of proof: (A) Each of the axioms (I-X) has the property E; (B) If U has the property E, then W/p' U has the property E; (C) If both U and U --+ V have the property E, then V has the property E; (D) Every thesis of sentential logic has the property E. Example 5. of definition? Example 6.

Would it be justified to introduce a corresponding method Show that the following expressiona are theses: (XI)

(XII) (XIII)

p --+ [(p --+ q) --+ {(p --+ q) --+ q}]; p --+ [(p --+ q) --+ q];

(XIV)

[p --+ (q --+ r)] --+ [q --+ (p --+ r)];

(XV)

(q --+ r) --+ [(p --+ q) -+ (p --+ r)];

(XVI)

(XVII)

74.

p --+ P

p --+ (p

(p --+ p)

--+ q); --+ p,

PROOF FROM ASSUMPTIONS

Let a be any set of formulas of sentential logic. Then we denote as ~(a), the intersection of all sets b which satisfy the following conditions:

PROOF FROM ASSUMPTIONS

205

(i) a is included in b ; (ij) every thesis is contained in b ; (iij) if both U and U --+ V are in b, then V is in b [in other words: the set (t(a) is obtained from a, by first adding all theses of sentential logic, and then applying again and again the rule of detachment or modus ponens (T 3)].

r-

Whenever U belongs to (t(a), we write a U, and we say that U is derivable from the set a of assumptions by means of sentential logic. It may happen that the set of assumptions is the empty set 0; in that case, we have (t(0)=%, where % is the set of all theses. It is clearly justified to apply the following method of proof (which is known as proof by recursion on the derivation of a formula U from the set a of assumptions): (A) (B) (C)

Every formula in a has the property E; Every thesis has the property E; If both U and U --+ V have the property E, then V has also the property E;

(D)

Every formula U in (t(a) has the property E.

(1) Using this method of proof, it is very easy to establish the following results;

r-

We have a U, if and only if, for some finite subset aO of a, we have a°r- U; For every expression U in (t(a), we can draw a pedigree which show; why U must be in (t(a). This pedigree is called the (formal) derivation of U from a (by means of sentential logic). The pedigree ends with U and for each application of the above condition (iij) [or modus ponens] it contains a formula Y which is connected with two formulas X and X --+ Y. The "branches" in the pedigree start from expressions which either are in a or are theses. X

X--+Y

Y

U

206

SURVEY OF SYMBOLIC LOGIC

(2) It follows that:

An expression U is in ~(a) if and only if there is a finite set b, containing only expressions which either are in a or are theses, and from which U can be obtained by means of modus ponens. (3) Let b be any set of expressions; then by MP(b), we shall denote the set of all expressions obtainable from b by means of modus ponens. (4) In order to show that every expression U in MP(b) has a certain property E, we may clearly apply the following method of proof: (B) (C)

Every thesis in b has the property E; Every expression in b, which is not a thesis, has the property FJ; If both Sand S --+ T have the property E, then T also has the property E;

(D)

Every expression U in MP(b) has the property E.

(A)

(5) Using the method of proof described under (4), we now establish the following statement:

Whenever U is contained in MP(b), we can find in b a finite number of expressions AI' A 2 , Aa, ... , A k , which are not theses, and such that:

(I) is a thesis of sentential logic. Proof. The property E to be considered will be, of course, the existence of a thesis (I), for a suitable number k,

ad (A)

If U is a thesis, we can take k = O.

ad (B) If A is in b, we can take k= I; for Al --+ A, where Al is the same formula as A, is a thesis, by the result under XI of Example 6 in Section 73. ad (C)

Suppose we have theses Sm, or:

and X n , or: G1--+ ( ... --+ (G n --+ (S --+T)) ... ),

PROOF FROM ASSUMPTIONS

207

where B ll B 2 , ••• , B m , Cll ... , C; are formulas in b which are not theses. We wish to show that T mn' or: B 1 --+ (B 2 --+ ( ... --+ (B m --+ (C1 --+ ( ••• --+ (Cn --+ T) ... ))) ... )),

a thesis. It will be clearly sufficient to show that, for every choice of the formulas S, T, B 1 , B 2 , ..• , B m , C1 , ••• , the formula: IS

en,

s; --+ rx, --+ T mn) is a thesis. We apply proof by recursion, first on n and then on m. (i) So --+ (X o --+ Too) is the formula S --+ [(S --+ T) --+ T], which is a thesis by the result under (XIII) of the above-mentioned Example. (ij) Recursion on n will be justified, if we show that: [So --+ (X q_1 --+ T Oq-1)] --+ [So --+ (X q --+ TOq)], where X q is C1 --+ X q_1 and T Oq is C1 --+ T Oq-1' is a thesis; and indeed, by the result under (XV), twice applied, the formula: [So --+ (X q_1 --+ TOq-l)]

--+

[So --+ {(C 1 --+ X q_1) --+ (C1 --+ TOq_1)}]

must be a thesis. (iij) Recursion on m will be justified, if we show that: [Sp-1

--+ (X n --+

T p-1n)] --+ [Sp --+ (X n --+ T pn)],

where Sp is B 1 --+ Sp_1 and T pn is B 1 --+ T p-1n, is a thesis. So we have to show that: [Sp_1 --+ (X n --+ T p-1n)] --+ [(B 1 --+ Sp_1)

--+ {X n --+ (B1 -+ T p-1n)}]

is a thesis. We first show, applying the same method as under (ij), that: [Sp_1 --+ (X n --+ T p-1n)] --+ [(B 1 --+ SV-1) --+ {B1 --+ (X n --+ TV_In)}] is a thesis. Then we apply the result under (XIV), and finally again the result under (XV). - This terminates the proof of theorem (5). (6)

We now prove the Deduction Theorem:

We have a ~ U, if and only if AI' A 2 , A a, ... , A k , such that:

a contains a finite number of formulas (I)

tS

a thesis of sentential logic.

208

SURVEY OF SYMBOLIC LOGIC

Proof. Clearly, the existence of a thesis (I) implies a I- U. - In order to prove the converse statement we consider the set b of theorem (2). Now U is in MP(b), so by theorem (4) a thesis (I) can be found. As AI' A 2 , A a, ... , A k are in b, but are not theses, it follows that they must be in a. Although established by Tarski as far back as 1921, the deduction theorem was published only in 1929, independently, by Herbrand and Tarski. It is known to hold for all logical systems in which, besides modus ponens and the rule of substitution, we have the following theses: p~

[p

~

(q ~ r)]

~

(q~p),

[(p

~

q)

~

(p

~

(7) We say that a set a is incon.sistent, if ~(a) set of all formulas of sentential logic. (8) A set a is inconsistent, if and only if formulas V and V.

r)]. =~,

~(a)

where

~

is the

contains two

Proof. We have only to show that, if ~(a) contains two formulas V and V, it contains every formula U. Now by the result under (XVI) of Example 6 in the previous Section, V ~ (V ~ U) is a thesis of sentential logic; it follows that U is contained in ~(a). (9) A set a is inconsistent, if and only if some finite subset aD of a is inconsistent. This follows at once from the theorems under (1) and (8). contains some atom p which (10) A set a is inconsistent, if ~(a) does not appear in any formula of a. Proof. Suppose that ~(a) contains the atom p which does not appear in any formula of a. By the deduction theorem, a must contain formulas AI' A 2 , ••• , A k , such that:

Al

~

(A 2 ~

( •••

~

(A k ~ p) ...

»

is a thesis; now let U be an arbitrary formula. By the rule of substitution, the formula:

Al

~

(A 2 ~

( •••

~

(A k ~ U) ... J)

is a thesis; it follows that U is in ~(a). (11) We have a f- U, if and only if au {U} is inconsistent. Proof.

Suppose that a u {U} is inconsistent. Then a u {U}

I-

U,

209

PROOF FROM ASSUMPTIONS

so we can find formulas

AI'

A 2,

••• ,

A k in

0

u {V}, such that:

Al -+ (A 2 -+ ( ... -+ (A k -+ U) ... )) is a thesis. Now if V does not happen to appear among AI> A 2 , ... , A k , then it follows that all formulas Ai must be in 0, and so 0 f- U. Otherwise, some formulas Ai will be in 0, and the others must be occurrences of V; let C;+l' ... , Ck be the occurrences of V, and let B l , .•• , B; be the remaining formulas Ai' By applying the results under (XIV) and (XV) of Example 6 in the previous Section, we show that: B l -+ ( ... -+ (B; -+ (O;+l -+ ( ... -+ (Ck -+ U) ... ))) ... )

is a thesis. As B l ,

... ,

B; are in

it follows that:

0,

Ci+l -+ ( ... -+ (Ck -+ U) ... ),

or:

V -+

( ... -+

(V -+ U) ... ),

is in ~(o). By applying Axiom (II), we find that V -+ U is in ~(o); so, by the result under (XVII) of the above-mentioned Example, U must be in ~(O). (12) If the set 0 is consistent, then either 0 must be consistent. This follows immediately from theorem (II). c u {U} and 0 U {V} can be consistent.

U

{U} or

0 U

{V}

Notice that both

(13) If o=~(o), then the set 0 is said to be a (deductive) system (of sentential logic). Every set ~(o) is a deductive system. If 0 and b are deductive systems such that 0 ~ b, then b is said to be an extension of u. A deductive system 0 is said to be complete, if 0 contains (at least) one out of every two formulas U and V. (14) Every consistent system extension (A. Lindenbaum). Proof.

0

has a complete and consistent

Let:

be an enumeration of all formulas of sententiallogio. Now we construct a complete extension of 0, as follows. If 0 u {Ul} is consistent, then 0 1 = ~(o u {Ul } ) ; otherwise, 0 1= 0= ~(o U {VI}); ... ; if 0k-l U {Uk} is

210

SURVEY OF SYMBOLIC LOGIC

consistent, then ak=[(ak-l U {Uk}); otherwise ... ; and so on, ad into It will be clear that the union of all systems a, aI' ~, ... , ak' ... is a complete and consistent extension of a. (15) Every deductive system is the intersection of all its complete extensions (Tarski 1936; ct. Section 88, Example 1). In particular, the set '1: of all theses is the intersection of all complete systems. (16) Every function w, which assigns to each formula U a value w( U) = 2 or = 0, in accordance with the following conditions:

(i)

w(X)= 2, if and only if w(X)=O;

(ij) w(X (iij) w(X

--+ Y) = 0, ~

if and only if w(X) = 2 and w( Y) = 0;

Y)=2, if and only if w(X)=w(Y);

(iv) w(XvY)=w(X ~ Y);w(X& Y)=w(X"vY);w(X/Y)=w(XvY); is called a valuation. - It will be clear that a valuation is uniquely determined by the values which it assigns to the atoms. (17)

A formula U is called a logical identity if, for every valuation = 2.

w, we have w( U)

(18)

Every thesis is a logical identity.

Proof.

We apply recursion on the derivation of a thesis.

It is easy to see that all formulas (I)-(X) under (T 1) are logical identities. This part of the proof is most conveniently carried out by the method of semantic tableaux (ct. Section 70). ad (A)

ad (B) Suppose that U is a logical identity, and let w, W, p be, respectively, an arbitrary valuation, an arbitrary formula, and an arbitrary atom; we wish to show that w( W /p. U) = 2. Let us introduce a valuation w*, as follows: w*(p)=w(W), and w*(q)=w(q) for every atom q different from p; then it is easy to show, by recursion on the construction of a formula X, that w(W/p·X)=w*(X). It follows that, in particular, w( W /P: U) = w* (U) = 2. ad (C) If w(U --+ V) = 2 and w(U) = 2, then we cannot have, by rule (ij) under (16), w( V) = 0, hence w( V) = 2. So, if both U --+ V and U are logical identities, then V is also a logical identity.

(19) There is a one-to-one correspondence between valuations w and complete and consistent systems a such that, whenever this correspondence holds between wand a, we have, for every formula U, w( U) = 2, if and only if U is in a.

PROOF FROM ASSUMPTIONS

sn

Proof. There is clearly a one-to-one correspondence between arbitrary sets 0 of formulas and arbitrary functions w such that, for every formula U, we have w( U) = 2 or = 0, according as U is or is not in a. We now show that w is a valuation if and only if the corresponding set 0 is a complete and consistent system. (I) Suppose that w is a valuation. We have to show that, for the formulas U in (£(0), w( U) = 2; it then follows that (£(0) = c. We apply recursion on the derivation of U. ad (A) For every formula U in 0, we have w( U) = 2. ad (B) For every thesis U, we have w(U)=2 by theorem (I8). ad (C) As shown in the proof of theorem (18), if w(U ~ V) = 2 and w( U) = 2, then w( V) = 2. (II) Suppose that 0 is a complete and consistent system. We have to show that w satisfies the conditions (i)-(iv) in definition (16). ad (i) This condition is clearly satisfied. ad (ij) First suppose that w(X)=O. Then X is not in 0, hence X is in 0, and, as by the result in Section 73, Example 6 under (XVI), the formula X ~ (X ~ Y) is a thesis, it follows that X ->- Y is in 0; therefore, w(X ~ Y) = 2. Suppose now that w(Y) = 2. Then Y is in 0 and, as Y ~ (X ~ Y) is a thesis by axiom (I), X ~ Y must be in 0; therefore, w(X ~ Y) = 2. Finally, suppose that w(X)=2, w(Y)=O. Then both X and Yare in 0, and, if X ~ Y were in 0, then Y would be in 0 and 0 would be inconsistent. So X ~ Y cannot be in 0, and hence w(X ~ Y) = o. ad (iij)-(iv) These conditions are satisfied on account of axioms (IV)-(VI) and (VIII)-(X). (20)

We now obtain the completeness theorem for sentential logic:

Every logical identity is a thesis. Proof. If U is a logical identity, then by theorem (19) it is contained in every complete and consistent system. Being contained in the intersection of all complete systems it is, by theorem (15), a thesis. (21) For every formula U, we are in exactly one of the following three cases: (i) TJ is a logical identity and hence a thesis; (ij) fJ is a logical identity and hence a thesis;

212

SURVEY OF SYMBOLIC LOGIC

(iij) Neither U nor U is a logical identity and hence neither U nor U can be a thesis, and there is a decision procedure which enables us effectively to find out in which case we are for any given formula U. Proof. But for the last part, this is only a more elaborate statement of the result under (20). In order to find out whether or not a given formula U (or U) is a logical identity, we may use a semantic tableau (ct. Section 70), in which the only initial formula is U (or U) in the right column. (22) For every formula U and every set a, we are in exactly one of the following four cases: (i) There is no valuation w such that, for every formula A in a, we have w(A) = 2; in this case, a is inconsistent; (ij) There are valuations w such that, for every formula A in a, w(A)=2, and for any such valuation w(U)=2; in this case, we have a~ U, but not a~ U; (iij) There are valuations w such that, for every formula A in a, w(A)=2, and for any such valuation w(U)=O; in this case, we have a~ U, but not a~ U; (iv) There are valuations w such that, for every formula A in a, w(A)=2; for some valuations w of this kind, we have w(U)=2, and for others, we have w( U) = 0; in this case, we have neither a ~ U nor a~ U.

If a is a finite set, then there is a decision procedure which enables us, for a given formula U, effectively to find out in which case we are.

This theorem is proved in the same manner as theorem (21). (23) Suppose that A +i>- B is a thesis; then, for any choice of an expression U and of an atom p, the formula (A/p· U) +i>- (B/p. U) is a thesis. Proof. As A +i>- B is a thesis, it is a logical identity; so we have, for any valuation w, w(A +i>- B)=2, and hence w(A)=w(B). It follows, by the argument in the proof of theorem (18), ad (B), that, for every valuation w, we have w(A/p. U) =w(B/p. U). Therefore, the formula (A/p. U) +i>- (B/p· U) is a logical identity and thus, by theorem (20), it must be a thesis.

213

REVISION OF THE AXIOM SYSTEM

(24) Suppose that A *+ B is a thesis, let U be any expression, and let p be an atom; then Al-p- U is a thesis, if and only if Bl-p- U is a thesis. Example 1.

Give a detailed proof of the results under (I).

Example 2. Complete the proof of theorem (5) by giving a more detailed statement of the argument under (iij], Example 3. -Iuatify the remark which follows the above proof of the deduction theorem (this remark is due to A. Church). Example 4. Complete the proof of theorem (II) by giving a detailed statement of the application of the results under (XIV) and (XV). Example 5. Complete the proof of theorem (14) by showing that the union of all systems (I, (11' lit, ... is (i) a system, (ij) consistent, and (iij) complete. Example 6. Show that a valuation is uniquely determined by the values which it assigns to the atoms; moreover, show that for each choice of values for the atoms, a valuation is obtained. Example 7.

Give a proof of theorem (22).

Example 8.

Complete the proof of theorem (23).

75.

REVISION OF THE AXIOM SYSTEM - ELIMINATION OF THE RULE OF SUBSTITUTION

We now replace the rules (T 1-4), as stated in Section 73, by the following system. (T 1*) Let P, Q, and R be any expressions; then the expressions:

(I) (II)

P -+ (Q -+ P);

[P -+ (P -+ Q)J -+ (P -+ Q);

(III)

(P -+ Q) -+ [(Q -+ R)

(IV)

(P

*+

Q) -+ (P -+ Q);

(V)

(P

*+

Q) -+ (Q -+ P);

(VI)

(VII) (VIII) (IX)

(P -+ Q) -+ [(Q

(Q

P)

-+ 15) -+ (P -+ Q);

(P v Q)

*+

(P & Q)

(X) (P / Q) are theses;

--+

(15 -+ Q) ;

*+

*+

(15v Q);

(15v Q),

--+

(P -+ R)];

--+

(P

*+

Q)];

214

SURVEY OF SYMBOLIC LOGIC

(T 2*) If both U and U --)- V are theses, then V is also a thesis. (T 3*) Nothing is a thesis (of sentential logic), except on account of rules (T 1*-2*). (1) The axiom systems (T 1~4) and (T 1*-3*) are equivalent, that is, every formula which is a thesis on account of (T 1-4) is also a thesis on account of (T 1*-3*), and conversely.

It is easy to see that, if U is a thesis on account of 1*-3*), then U is also a thesis on account of (T 1-4). For replacing 1-4) by (T 1*-3*) may be construed as restricting the application the rule of substitution (T 2) to those cases, in which U is one

Proof. (T (T of of

the axioms. We now turn to proving that, if U is a thesis on account of (T 1-4), then U is also a thesis on account of (T 1*-3*). The method to be applied is known as the method of anticipating substitutions. We first show that, if in the derivation of a thesis U on the basis of (T 1-4) an application of rule (T 2) is preceded by an application of rule (T 3), then the relative order of the applications of these rules can be reversed. Let us consider the following fragment in the derivation of U: X X--)-Y Y A/p·Y

This fragment can be replaced by: X

X--)-Y

------'-

A/p·X -~_.

_ _ .. _------,-

A(p·(X --)- Y) ------------~

A/p·Y

For, on account of rule (S 3) in Section 73, the substitution result A(p·(X --)- Y) is (A(p·X) --)- (A/p. Y), and hence the second fragment can also be written as: X

A/p.X

(A/p. X) --)- (A(p. Y) --A/p.y

_._.,--_._._~-----_._-

This last fragment clearly consists of two applications of rule (T 2) followed by an application of rule (T 3).

ELEMENTARY LOGIC

215

Now let us consider the derivation of the thesis U as a whole (we could, instead, also apply the method of proof by recursiou on the derivaoion of a thesis). It will be clear that by repeatedly applying the trausformation which has just been described, we shall finally obtain a derivation in which first all applications of rule (T 2) are carried out, and afterwards only rule (T 3) is applied. Therefore. let us consider, as an example, axiom (I): P --+ (q --+ p);

what kind of formula will result if, starting from this axiom, a finite number of successive substitutions is carried out? On the basis of the above rules (8 1-4), it is easy to answer this question: the resulting formula will always retain the structure P --+ (Q --+ P); hence the result of any finite number of successive substitutions can also be obtained by one single substitution; and it can also be obtained on the basis of rule (T 1*). The same argumeut applies, of course, to the remaining axioms (II~X). This terminates our proof. (2) We now turn to the results in Section 74. Although, in proving these results, we had to refer to the system (T 1-3), their statement does not refer to this system. Hence, on account of the result under (1) in this Section, the results in Section 74 remain valid, if the word "thesis" is now interpreted as "thesis in accordance with rules (T 1*-3*)". Example 1. Prove that the result of a finite number of successive substitutions, starting from axiom (I) in the system (T 1-3) is always a formula of the form P -+ (Q -+ Pl. - Hint: first give a definition of the notion of a formula of the form P -+ (Q -+ P); then apply recursion on the number of successive substitutions. Example 2. Give a proof of the result under (2) in this Section which applies recursion on the derivation of a thesis. Example 3. Prove theorem (18) in Section 74 directly on the basis of the system (T 1*-3*).

76.

ELEMENTARY LOGIC

In order to construct elementary logic, we need the following atoms: Pl. P2' P3' P4' ... , Pk, ... ; sometimes, we write instead A, B. C, a1(x2), a l(x3), .... a 2(xl ) , a 2(x2)• . .. , a3(xl ) • .. . , ai(xk), ... , 1'I (XI, Xl)' r1(x V x 2), rl(x l, X3)• .. . , r l(x2, Xl)' r 1(x2, X2), ... , r l(x3, Xl).....

(JI(XI),

216 r 2(xl ,

SURVEY OF SYMBOLIC LOGIC Xl)'

SI(XI, ~l'

r 2(xl ,

X 2),

••• ,

r 2{x2 ,

Xl)' ••• ,

ri{x,."

Xl)' SI(XI, Xl' X 2), SI(Xl, Xl' Xa),

Xl)' ••• ;

... , Sl(Xl,

X 2, Xl)' SI{Xl> X 2, X 2), •.. ,

SI{XI, Xa, Xl)' .•• , Sl(X 2, Xl> Xl)' Sl(X2, Xl' X 2), .•• , Sl(X 2, X 2, Xl), ... , SI(Xa, Xl> Xl), ••• , Si(Xk, Xl' X m ) , ....

The symbols Xl> X 2, Xa, ... , X k, ••• are called individual variables; more often, we write instead X, y, z, . The symbols al> a2 , aa, ... , ai' are called one-place (or unary) predicate parameters; more often, we write instead a, b, c, ... or A,B,C, .... The symbols rl> r2 , .•• , r i, ... are called two-place (or binary) predicate parameters; more often, we write instead r, r', ... or A, B, .... The symbols Sl' ... , si' ... are called three-place (or ternary) predicate parameters; more often, we write instead s, ... or A, .... There is no objection to introducing four- and more-place predicates; however, it will not be necessary to mention them explicitly in the present context. In addition to the sentential connectives, we need quantifiers (Xl') (X 2), (xa), ... , (X k), ••• , and (Ex l ) , (Ex2 ) , (Exa), ... , (Ex k ) ..• ; both sentential connectives and quantifiers will be referred to as operators (of elementary logic). The notion of an expression (or formula) of elementary logic is defined much along the same lines as the notion of an expression of sentential logic; however, the necessity of avoiding confusion of free and bound variables creates certain complications. (1) In the remaining part of this Section we attach different meanings to the terms "expression" and "formula" (of elementary logic). We first define the notion of an expression, and then we introduce the more restricted notion of a formula. Later, only formulas will be taken into account, and then the terms can again be used indifferently. (F 1) Every atom is an expression; (F 2) If U is an expression, then rJ and, for k= 1,2, ... , (xk)U and (Exk)U are expressions; (F 3) If U and V are expressions, then U v V, U & V, U -)- V, U +* V, and U I V are expressions; (F 4) Such (individual) variables as appear in an atom are free in it;

ELEMENTARY LOGIC

217

If the variable X k appears in U, then it is bound in (x k ) U and in (Exk)U. Any other variables are free or bound in (xk)U and in (Exk)U, according as they are free or bound in U. Any variable is free or bound in D, according as it is free or bound in U; and free or bound in U V V, U & V, U -+ V, U ~ V, or U / V, according as it is free or bound in U or V (hence a variable may be both free and bound in one and the same expression; this is the confusion of free and bound variables which we wish to avoid); (F 5) The quantifiers (x k) and (Ex k) dominate U in (xk)U and (Exk)U, respectively. They dominate U in Y, (xv) V, or (Ex Q) V,

according as they dominate U in V; and they dominate U in Vv W, V & W, V -+ W, V ~ W, or V / W, according as they dominate U either in V or in W; (F 6) Nothing is an expression, nothing is free or bound in an expression, and nothing dominates an expression in an expression, except on account of rules (F 1-5); (F 7)

Every atom is a formula;

(F 8) If U is a formula, then D is a formula; (F 9) If U is a formula in which (Exk ) U are formulas;

Xk

is free, then both (xk)U and

(F 10) If both U and V are formulas, and if no variable is free in U and bound in V, or conversely, then U v V, U & V, U -+ V, U ~ V, and U / V are formulas; (F 11) If x, x', x", ... , y is any enumeration of all variables free in a formula U, then the formula (x)(x')(x") ... (y)U is a closure U' of U; if a formula U contains no free variables, then it is a sentence, and U is its own closure U'; (F 12) Nothing is a formula, the closure of a formula, or a sentence, except on account of rules (F 1-11). We now turn to introducing the notion of a thesis (of elementary logic). But it proves helpful first to introduce the notion of an application A of a set of formulas P of sentential logic, as follows. Let ql' qz, ... , q", be an enumeration of all atoms appearing in the formulas P, and let U l, U z, .. " U m be any expressions of elementary logic; then we take: (i) for A(qk)' the expression Uk(k= 1,2, ... , m); (ij) for A(Q), the expression A(Q); (iij) for A(Q V R), the expression

21H

SURVEY OF SYMBOLIC LOGIC

A(Q) V A(R), and likewise for A(Q & R), A(Q --+ R), A(Q ~ R), and A(Q / R). It will be clear that all expressions A(P) will be formulas, provided that the expressions U1> U 2 , ... , U m are formulas and that no variable is free in some of these expressions and bound in some others. (2) Now the notion of a thesis 01 elementary logic can be introduced as follows (v, v', v" may be any of the variables x, y, z, ... ). (T 1) Every formula, which is an application of one of the axioms (I~X), as stated in Section 72, under (T 1), is a thesis. (T 2) Every formula of one of the following kinds: (a)

(v)U(v) -+ U(v'),

(b)

U(v) -+ (Ev')U(v'),

(c)

(v)[V(v) -+ W] -+ {(Ev')V(v') -+ W},

(d)

(Ev)[(Ev')U(v') --+ U(v)],

(e)

(Ev)[U(v) -+ (v')U(v')].

is a thesis. (T 3) If both U and U -+ V are theses, then V is a thesis. (T 4) If U(v) is a thesis, then (v') V(v') is a thesis, provided v' either v or does not occur in U(v). (T 5) Nothing is a thesis, except by rules (T 1~4).

IS

In connection with these rules, the following remarks are in order. (3)

Every thesis of elementary logic is a formula.

(4) Every thesis of sentential logic is a thesis of elementary logic. This follows from the fact, that every thesis of sentential logic can be derived on the basis of the system (T 1*-3*), and that rule (T 1*) is a restricted version of rule (T 1) as above. Clearly, every application A(P) of a thesis P of sentential logic is a thesis of elementary logic; we shall say that A(P) is a thesis by sentential logic. Conversely, every formula of sentential logic, which is a thesis of elementary logic, is already a thesis of sentential logic.

(5) It is easy to see that, on account of rule (T 4), every closure of a thesis is again a thesis. (6)

If we introduce, as additional axioms, all formulas: (Ex)U(x) ~ (x)U(x),

then the above rules can be considerably simplified.

219

ELEMENTARY LOGIC

(7) In most presentations of elementary logic, the rules of derivation are affected with certain restrictive conditions. In the above rules, such restrictive conditions are tacitly implied by the stipulation that the resulting expressions must be formulas. (8) We are now in a position to deal with the notion of proof from assumptions in elementary logic. Let a be any set of formulas of elementary logic. Then we denote as
(10) The deduction. theorem (for elementary logic) can now be stated as follows:

Let a be any set of formulas and let U be any [ormula ; let a' and U' be the set of all closures of [ormulas in a and the closure of U, respectively,. then the following conditions are equivalent: (i)

U

(ij)

a contains a finite number of formulas AI' A 2 ,

E

Q:(a);

A; -+ (A; -+ ( ... -+ (A~ M

(iij)

... ,

A k , such that:

-+ U') ... ))

a thesis;

aff- U',

Proof. We first show, by recursion on the derivation, that (i) implies (ij). It will be clear that this implication holds true for every thesis and for every formula in a, and that it holds true for (x) V(x) whenever it holds true for V(x). Now we suppose that the implication .holds true for the formulas

220

SURVEY OF SYMBOLIC LOGIC

V and V -'>- W; we have to show that it also holds true for W. Let B 1 , B 2 , ••• , B m , C1 , ... , O; be formulas in a such that 8 m , or: B~

-'>-

(B~

-'>- ( ... --i>

(B~

--i>

V') ... »),

and X n , or: C~

--i> ( ... --i>

(C~

--i>

(V

--i>

W)') ... ),

are theses. By sentential logic, it follows that T mn' or: [( V

--i>

W)'

-'>-

{B~

--i> --i>

(V' (B~

--i>

W')]

--i>

--i> ( ••• --i>

(B~

--i>

(C;

-'>- ( ... --i>

(C~

--i>

W') ... ») ... ))},

is a thesis. And, as (V --i> W)' --i> (V' --i> W') is also a thesis, this terminates the first part of our proof. - Furthermore, (ij) clearly implies (iij), and it is easy to show that (iij) implies (i). Example 1.

Prove that every thesis of elementary logic is a formula.

Example 2. Prove that every formula of sentential logic, which is a thesis of elementary logic, is already a thesis of sentential logic. - Hint: in a derivation of such a thesis, we cancel all quantifiers, and we replace all individual variables by x; show that in this manner we obtain a derivation in sentential logic. Example 3. In connection with the statement under (6), find out which simplifications can be introduced. Example 4. In connection with the above proof of the deduction theorem, show that the implication of (ij) by (i) holds true for (x) V(x) whenever it holds true for V(x). Example 5.

Prove that every formula (V ---;. W)' ---;. (V' ---;. W') is a thesis.

- Of. Section 67, Example 2.

Example 6. Complete the above proof of the deduction theorem by showing that (iij) implies (i). Example 7. Is it true that, whenever U ---;. V(x) is a thesis, then U---;.(y)V(y) is also a thesis? Example 8. In our proof of the deduction theorem for elementary logic, we have introduced the notion of a formula being a "thesis by sentential logic" The introduction of this notion can be based on the definition: An expression of elementary logic will be called a sentential thesis, if it is both a. formula and an application of a thesis of sentential logic, and on the theorem:

Every sentential thesis is a theeie of elementary logic. Give a proof of this theorem and find out how it is involved in our proof of the deduction theorem.

INDIVIDUAL PARAMETERS, IDENTITY, FUNCTION PARAMETERS

221

Example 9. In the construction of a deductive theory we do not, usually, deduce all its theorems directly from the axioms. We add each theorem, once it has been deduced, to the axioms, and in deducing further theorems we use both the axioms and all theorems previously obtained. In order to reconcile this current practice with our above approach, prove the following theorem:

If a I-Xl' a U {Xl} I-X., a U {Xl> X.} I- X., ... , a U {Xl' X., X., ... , Xk-d I-X k, then a I- X k • - Hint: it is sufficient to show: If a I- X and a U {X} I- Y. then a I- Y. 77.

INDIVIDUAL

PARAMETERS,

IDENTITY,

FUNCTION

PARAMETERS

The following amplifications of elementary logic have proved helpful both in view of establishing certain general results about this logical system and in connection with its application as a basis for the formalisation of certain mathematical theories. (1) In addition to the individual variables we often use individual parameters (usually called individual constants) i, j, k, ... or iI' i 2, ... ... i k , ••• ; these parameters differ from the variables by the absence of corresponding quantifiers; accordingly, the distinction between free and bound variables is not extended to the individual parameters. On the other hand, an amplification of (T 2) is in order, permitting us to replace the variables v' in (a) and v in (b) by any individual parameter i, j, k, .... (2) We introduce atoms Xk=X I , xk=i j , ik=x l , ik=i l which are treated in the same manner as r(x k, Xl)' r(x k, ill, 1'(i k, Xl)' r(x k, Xl)' The following clause must be added to (T 2): Every formula: v=v

(a)

and every formula: (b)

v=v'

--+

[U(v)

--+

U(v')j,

is a thesis. The formulas under (a) and (b) are denoted as axioms identity (or 01 equality), and the amplified version of elementary logic which results is denoted as elementary logic with identity (or with equality). - It is sometimes convenient to write v ¥= v' instead of v=v' and to treat v¥=v' as an atom. (3) Finally, we introduce lunction parameters I, g, ... which stand for arbitrary functions of one or two (sometimes three or more) variables; we shall sometimes also say that these parameters stand for unary or binary (ternary, ... ) operations.

01

222

SURVEY OF SYMBOLIC LOGIC

To fix the ideas, suppose that f and g stand, respectively, for a unary and for a binary operation. We introduce terms, as follows: (i) every individual variable and every individual parameter is a term; (ij) if t is a term, then f(t) is also a term; (iij) if t and t' are terms, then g(t, t') is also a term; (iv) nothing is a term, except on account of rules (i)-(iij). Now to the atoms introduced in Section 76, we add: a(t), b(t), c(t),

.

r(t, t'), r'(t, t'),

.

s(t, t', t"), ...

for arbitrary terms t, r, t"; in such an atom, every individual variable which appears, is free. To rule (T 2), we add the clause: All formulas: (fl)

(x)(Ey)[f(x)=y],

(f2)

(x)(x'}(y)(y')[{f(x)=y &f(x')=y' &x=x'}-+y=y'],

(g 1) (x)(y)(Ez)[g(x, y)=z], (g 2)

(x)(x')(y)(y')(z)(z')[{g(x, y)=z & g(x', y')=z' & x=x' & & y = y'} -+ Z = z'] ,

(a')

(v)U(v) -+ U(t),

(b')

U(t) -+ (Ev)U(v),

are theses. The formulas under (f 1-2) are denoted as functionality axioms for f, and the formulas under (g 1-2), as functionality axioms for g; the changes which are required if more or different operation parameters appear are obvious. The formal system which finally results from the amplifications under (1)-(3) is currently denoted as elementary logic with identity and function symbols. Its properties are closely related to those of the simpler version of elementary logic which has been discussed in Section 76; so closely, in fact, that it is not necessary now to go more deeply into this matter. Example 1.

Prove that the forrnules : (x)(y)[x = y

--7

and (x)(y)(z)[(x = Y & Y

are theses.

Y

=

=

z)

x] --7

X

= zJ

HILBERT'S e-SYl\IBOL

223

Example 2. Show that the results of Section 76 carryover to elementary logic with identity.

78.

HILBERT'S e-SYMBOL

We now give a sketch of an alternative treatment of elementary logic which has been described in more detail by D. Hilbert and P. Bernays (1938). Instead of quantifiers, we introduce prefixes ez ' ell' e., ... ; if U(x) is an expression which contains the free variable x (and, perhaps, still other variables, free or bound), then we introduce a term e",U(x). With respect to the construction of new expressions, ezU(x) is treated as t(y, z, ... ) if U(x) contains additional free variables y, z, ... , and as an individual parameter if U(x) contains no such free variables. The variable x in e",U(x) is considered as bound, whereas other free variables appearing in U(x) are not bound bye",; an expression in which no variable is both free and bound is a formula. Then (T 2) is replaced by: (T 20) Every formula: U(y) -+ U(e",U(x»

is a thesis (weak e-axiom). The quantifiers can be defined in terms of the s-symbol, as follows: (Ex)U(x) __ U(e",U(x»; (x)U(x) __ U(e",U(x»;

starting from these definitions we can prove that Hilbert's version of elementary logic is equivalent to the customary version. The main advantage of Hilbert's version of elementary logic is that it narrows the gap between sentential logic and quantification theory; in particular, it enables us to reduce derivations in elementary logic to derivations in sentential logic. Specifically, Hilbert's first s-theorem states that any bound variable which appears in a derivation but which appears neither in the premisses nor in the conclusion can be eliminated. It follows from our discussion in Chapter 8 .that an equivalent result can also be obtained by different methods; this subject will be treated more thoroughly in Chapters 10 and II. Hilbert's treatment of quantification theory is also interesting in connection with the historical discussion in Section 5. For the absolute entity a corresponding to a property A can now be identified with exA(x). Then Plato's difficulties are seen to arise from his tacit sup-

224

SURVEY OF SYMBOLIC LOGIC

position that we must always have A(ExA(x)). According to Hilbert, however, we have A(ExA(x)) only if (Ex)A(x). On the other hand, it follows from Bolzano's analysis that there is no real difficulty in supposing that ExA(x) is a corruptible entity. Hence, if A is a property to be predicated of corruptible entities, then we have: (i) if some corruptible entity x has the property A, then (Ex)A(x) and hence A (ExA(x»), where ExA(x) is some corruptible entity; and (ij) if no -----corruptible entity has the property A, then (x)A(x), hence A(ExA(x)). Aristotle's opinion, according to which a universal statement (x)[A(x) _ B(x)) cannot be true unless we have, for some x, A(x), and hence (Ex)A(x), fits in with this interpretation. For obviously Aristotle supposes that, at any rate, we have A(ExA(x)) and, therefore, (Ex)A(x). Exam pIe.

79.

Show that the above definitions of the quantifiers are adequate.

LOGIC OF HIGHER ORDER

In spite of the various amplifications, described in Section 77, elementary logic does not always provide us with the apparatus which we need. Its deficiencies mainly derive from the fact that all its variables range over the same domain and that, accordingly, all quantifiers refer to one and the same "universe" (ct. Section 63). In many theories of mathematics we find quantifications which refer to more than one universe, and in such cases a straightforward formalisation would require various kinds of variables, the variables of each kind ranging over one specific universe. It will be useful to mention a few examples of such a situation. (1) In elementary geometry we find such expressions as "tor every point", "there is a line", "tor any circle", and so on, which suggest the necessity of point variables P, Q, R, ... , line variables l, m, n, ... , variables I', ,1, E, ... for circles, and so on (in point of fact, such variables were already used by Greek mathematicians); in a formalisation of geometry it would be natural to use such variables and to introduce corresponding quantifiers. In this case, however, we could also use only one kind of variables, ranging over arbitrary figures, provided we introduced certain predicates P, L, C, ... in order to be able to express the supposition that a figure x is a point, a line, a circle, ... ; in this case, one kind of quantifier would do.

LOGIC OF HIGHER ORDER

225

(2) In modern algebra (cl. Section 54) we often say: "there is a number", "lor any ideal". Here again, it would be natural to use variables x, y, z, ... for numbers and variables X, L, ... for ideals; in a formalisation of algebra, we would, accordingly, introduce two different kinds of quantifiers. (3) In a formalisation of Dedekind's theory, we would need variables x, y, z, ... for natural numbers, variables X, L, M, ... for sets, and variables I, g, h, ... for functions. Now we could meet with this demand of applied logic by extending elementary logic into a similar system in which, however, the presence of various kinds of quantifiers was taken into account; the construction of such a "many-sorted" version of elementary logic presents no difficulty. In constructing an axiom system for Dedekind's theory of natural numbers, however, it would be necessary to state certain axioms concerning functions and sets which, in our previous exposition, we took for granted, for instance: (X)(f)(EY)(y)[Y(y) -

(Ex){X(x) & y=/(x)}].

However, this approach turns out to be not the most efficient one. For these additional axioms are by no means typical of Dedekind's theory; they appear in every deductive theory in which, likewise, we apply the notions of an arbitrary function and of an arbitrary set. Therefore, it is more natural to consider such axioms as belonging, not to the particular disciplines in which they are applied, but to logic. Now if we try to carry out the construction of a logical system in accordance with this conception, we find that the demands of the various deductive theories are strongly divergent and. so to speak, unbounded. In topology, for instance, we need variables (and corresponding quantifiers) for points, for point sets, for families of point sets (it may happen that we have to deal with the notion of an arbitrary neighbourhood system in a given Hausdorff space) and for point-to-point functions (or mappings); if we have to deal with sequences of points or of sequences of point sets, then we need, in addition, variables for functions of points or point sets to natural numbers, and hence for natural numbers as well (and again we need quantifiers for each kind of variable); if we consider a certain family of Hausdorff spaces, then the situation may become even more involved.

226

SURVEY OF SYMBOLIC LOGIC

The variety of variables and quantifiers which we need in a logical system intended to provide an apparatus for the formalisation of any deductive discipline whatsoever naturally creates the demand for a certain economy. The answer to this demand is the logic of higher order based upon the theory of types. This system has originated from the system of Principia M athematica and detailed descriptions of it have been given by R. Carnap (1929, 1954), A. Tarski (1933), and A. Church (1940). I shall now give a sketch of this system, which is based on its treatment by Tarski and Church. (i) We have variables for individuals xo, Yo, zo, ... , vo, ... ; variables for sets Xl>YI'~'" " VI' •.• ; variables for families (ofsets) X2'Y2'~' ..• , V 2, ... ; variables for classes (of families) X a, Ya, Za, ... , Va' ... ; .•. ; variables for classes of type k: x k, Yk, Zk' ... , V k, ... ; .... (ij)

We have atoms x1(Xo), x1(Yo), x1(zo), ... , YI(XO)' YI(YO)' ... ,

~(xo)'

... ; X 2(X1), X 2(YI)' ... , Y2(X 1), ... ; Z2(X 1), ... ; x a(x2), .•. ; ... ; Vk(V k-I), ... ;

roughly speaking, an atom Vk(V k_l) expresses the condition that the class V k of type k contains the class V k-l of type k - 1 as an element. (iij) The notion of a formula is defined as in Section 76. (iv) As to sentential logic and to the theory of quantification, we adopt the treatment in Sections 73-76. In the rules (T 2) and (T 4) the variables V and v' must, of course, be of the same type k.

In accordance with the above remarks, we now introduce a number of additional axioms. (v) Let U(v k ) be a formula in which the variable vk+l does not occur; then the closure of the formula:

is an axiom. These axioms will be referred to as reducibility axioms, on account of their connection with Russell's axiom of reducibility; S. Lesniewski has proposed the name of pseudo-definitions. (vi) The notion of identity (or equality) can be defined by stating, for k=O, 1, 2, .. .:

On the basis of this definition (which goes back to Leibniz), wo can state the extensionality axioms as follows: Vk+l =Wk+l

-<+ (Xk)[Vk+l(X k) -<+ Wk+l(X k)),

LOGIC OF HIGHER ORDER

227

These axioms imply the uniqueness of the vk+l whose existence is required by the reducibility axioms. (vij) It has been shown by N. Wiener (1914) and by K. Kuratowski (1921) that in a system of higher-order logic based on the theory of types (and in related systems to be discussed later) we can define the notion of an ordered pair, and hence the notion of a relation. Let {x k, Y~} be the Zk+l such that: (Zk)[Zk+l(Zk) ++ (Zk=X k v Zk=Yk)]

[its existence and uniqueness are guaranteed by the axioms under (v) and (vi)], and let {xd be {x k , x k } ; then the ordered pair can be defined as {{x k } , {xk , yd}. And a binary relation can be defined as a class all elements of which are ordered pairs. This treatment can obviously be extended so as to cover both three-term, four-term, ... relations and heterogeneous relations (that is, relations whose terms are of different type). And, of course, the notion of a function (of one variable, or of two of more variables, the function and the argument variables being of any type) is also definable. Moreover, the reducibility and extensionality axioms entail the existence and uniqueness of functions and relations in all those cases in which it is currently accepted as a matter of course. (viij) The axiom of choice must also be stated for each type separately. One method is first to extend Hilbert's s-notation to the logic of higher order and then to state the strong e-axioms:

(ix) Finally, it is usual to include an axiom of infinity. It is better, however, to postpone a discussion of this matter. In the first place, it seems doubtful whether such an axiom can be awarded the status of an axiom of pure logic; but, moreover (and this is more important), the statement of such an axiom presupposes an analysis of the notion of infinity which would be out of place in the present context. According to the result of this analysis, various statements of an axiom of infinity have been given; and these statements are by no means equivalent. (x) It is important to notice that the Deduction Theorem (cf. Section 76) carries over to the logic of higher order. For, first, it will

228

SURVEY OF SYMBOLIC LOGIC

be clear that it holds if the axioms (v)-(ix) are not added. Secondly, suppose we have U f- U in the higher order logic with additional axioms; let b be the set U to which are added those special cases of the additional axioms which playa role in the derivation of U from u. It will be clear that we have b f- U in the higher order logic without additional axioms; hence, by our first remark we can find some thesis: B I --+ (B 2 --+ ( ... --+ (Bm --+ (AI --+ ( ... --+ (An --+ U) ... ») ...»,

of the higher order logic without additional axioms, where B I , B 2 , ••• , B m are the above-mentioned special cases of the additional axioms, and AI' ... , An are in u. By modus ponens, it follows that Al --+ ( ... --+ --+ (An --+ U) ... ) is a thesis of the higher order logic with additional axioms. It will be clear that, into the logic of higher order, we can introduce parameters, provided they are subjected to the same type hierarchy as variables. For various reasons, it is not always convenient to introduce the complete apparatus of higher order as described in this Section. In such cases, we usually apply some intermediate system; in a logic .oi fourth order, for instance, we have individual variables x, y, z, ... , set variables X, Y, Z, ... , variables for families I, ID, 3, ..., and variables for classes $, 'fJ, C, .... It may then happen, however, that we can no longer rely upon the method of Wiener and Kuratowski. If this situation arises, then we must state certain axioms which guarantee the existence and uniqueness of such relations and functions as we need. Example 1. In the above exposition, we have spoken both of classes and of variables of a given type k; however, this terminology may lead us to neglect the distinction between use and mention of symbols (ct. Section 88). Therefore, it is often better to refer to individuals or classes as entities ot type 0, 1,2, .. , and to denote the corresponding variables (and parameters) as B1JrrWols oi category 0, 1,2, .... Find out which part of the above considerations then ought to be restated. Example 2. Give a formal definition of the notion of a binary relation, both terms of which are individuals. Example 3. Show that the higher order logic without additional axioms can be equivalently replaced by a system of natural deduction as discussed in Sections 67-70.

THE IDEA OF A "LOGICA MAGNA" -

80.

THE RELATIVITY OF LOGIC

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THE IDEA OF A "LOGICA MAGNA" - THE RELATIVITY OF LOGIC

Now suppose we wish to construct a certain mathematical theory on the basis of the logic of higher order. Then we would be naturally inclined to assume a certain specific axiom system a in such a manner that C£(a) coincides with the set of all theorems of the theory under consideration. However, it was already anticipated by Leibniz and more firmly established by Frege that, on the basis of a logical system of the above kind, pure arithmetic can be developed without an appeal to any specific axioms whatsoever; and this result was extended by Russell to the totality of pure mathematics. From our present point of view, this insight may not constitute such a fascinating discovery as it appeared to Leibniz, Frege, Russell, and their contemporaries. For we have deliberately incorporated into the logic of higher order a number of additional axioms, several of which are of a strongly existential nature, whereas some even display a definitely mathematical character. But for Frege, and also for Russell when he started writing his Principles of Mathematics, the situation was completely different; for these authors did not apply a type hierarchy, and they (more or less tacitly) assumed a naive Comprehension Axiom which, in the present context, can be stated as follows: (i) Objects which have a certain property in common constitute a class, which is defined by that property and of which those objects are the members; (ij) Classes are objects and hence they may in turn appear as members of a class; (iij) Classes which contain the same members are considered as identical; hence a class is uniquely determined by its members. Frege's deduction of arithmetic from logic and the difficulties which sprung from it (and which prompted Russell to include an embryonic version of the theory of types already in his Principles) will be discussed in Chapters 13 and 17. But I wish now to quote a few sentences from the Principles which contain a clear statement of the three parts of the Comprehension Axiom and which show that Russell was fully aware of its dangerous character. The reason that a contradiction [namely, the so-called RUBB811 paradox] emerges here is that we have taken it lIB an axiom that any propositional function containing only one variable is equivalent to essertdng membership of a class defined by the propositional function [part (il]. Either this axiom, or the principle that every class can be taken lIB one term (part (ij)], is plainly false ... (l.c. pp. 102/3) ... if .fa; and 'PX are equivalent propositions for all values

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of x, then the class of x's such that q,x is true is identical with the class of x's such that IpX is true [part (iij) ] (l.c. p. 20).

The above logic of higher order is the final result of the development of Russell's conception. A logical system which enables us to deduce the totality of pure mathematics without an appeal to any specific axioms is sometimes called a "Logica Magna" (or a "Grand Logic"; as far as I know, this notion was first used by W. Ackermann in 1937). At present, various systems of this kind are known. I mention the system of Principia Mathematica (1910-13) and its revisions by Carnap, Church, Tarski, and others; the system of S. Lesniewski (1927-31); the systems of Quine's New Foundations (1937; applied by J. Barkley Rosser, 1953) and Mathematical Logic (1940; revised version, 1951); and Skolem's formalisations of the systems of set theory of Zermelo (1922), Fraenkel (1929), and von Neumann (1938). These logical systems differ strongly both in their general appearance and as to their respective logical force, but they also share a number of properties. For instance, they all include the ordinary sentential calculus and the ordinary logic of quantification and hence for each of them we can prove an analogue to the completeness theorem of Lowenheim-Skolem-Godel, On the other hand, none of them can fully live up to the ideal which underlies their construction. For to each of the above systems (except, perhaps, the system of Lesniewski) we can apply the theorems of Godel and Rosser; hence, if such a system is consistent, it cannot afford a proof of a certain arithmetical theorem which, in a sense, expresses the consistency of that system. The variety of extant logical systems is still enhanced on account of the existence of other systems of logic which diverge already on the level of the sentential calculus. Among these systems, Heyting's intuitionistic logic must be considered as a Logica Magna, as it is meant to incorporate the totality of intuitionistic mathematics. On the basis of the modal and the many-valued sentential calculi of Lewis and of Post and Lukasiewicz, systems of quantificationallogic have been constructed by Ruth C. Barcan (1946) and R. Carnap (1946) and by Rosser and Turquette (1948, 1951, 1952), respectively; a first start towards developing a rudimentary version of set theory on the basis of many valued logic was made by Tameharu Shirai (1937) and Moh Shaw-Kwei (1954). At the Fourth Austrian Mathematics Congress (Vienna 1956) Skolem presented a construction of

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arithmetic within a system of set theory based upon a logic with infinitely many values. These facts have, naturally, lent strong support to the thesis of the relativity 0/ logic which has been defended, among others, by C. I. Lewis (1932), H. Hahn (1933), R. Carnap (1934), and L. Rougier (1939). According to the conventionalistic views of its adherents, the choice of a logical system depends upon the choice of the language which we want to use, and we are free to use that language which, for one reason or another, we consider as the most convenient one. For instance, one logician may prefer to speak in terms of Quine's New Foundations, whereas another considers the terminology of Skolem's formalisation of Fraenkel's system as more appropriate. Though the thesis of the relativity of logic certainly has the merit of providing a counter-weight to the dogmatic attitude sometimes displayed by the representatives of certain logical schools, it ought, on the other hand, not to be over-emphasised. At present. the various languages of modal and of many-valued logic are merely 'studied. they are hardly ever actually used. The language of intuitionistic logic and mathematics is confined to the members of a certain school, and they certainly do not prefer it for reasons of convenience alone. And, finally, the various "classical" languages must rather be considered as dialects of one and the same mother-language; each of them is able to express the (quasi-) totality of current mathematics, and if we compare the manner in which a given mathematical theory presents itself in each of them, then, in spite of certain peculiar features, we find in many respects a striking similarity. On the other hand, if we prefer Quine's system to Fraenkel-Skolem's, or vice versa, this is by no means a matter of mere convenience or personal taste; for Quine's system is weaker than Fraenkel-Skolem's, and hence its consistency can be established by more elementary methods; on the other hand, Fraenkel-Skolem's system provides proofs of certain theorems which in Quine's system are unprovable. Example. Find out which of the "additional axioms", as stated in Section 79, corresponds to each of the parts (i)-(iij) of the above Comprehension Axiom.

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BmLIOGRAPHICAL NOTES Somewhat different descriptions of elementary logic can be found in HILBEBT-AcKERMANN [1], lIERMES-SCHOLZ [1], KLEENE [7], CHURCH [6]. On the e-symbol, compare HILBEBT-BEBNAYS [1]. Systems of higher-order logic based on the theory of types are described by TARSKI [11] and CHuRCH [6]; their metamathematics has been studied recently by FITCH [1], COP! [1], LORENZEN [1], GANDY [1]. On many-valued logic, I recommend: LUKASmWIcz [1], RosSER-TuRQUETrE [1], DREBEN [2]. The problem of a Iogiea magna is discussed by ACKEBMANN [1], CHURCH [2], WANG [3]. The relativity of logic is defended by CABNAP [1], ROUGmR [2].