a definition of negation in extended basic logic

A Definition of Negation in Extended Basic LogicAuthor(s): Frederic B. FitchSource: The Journal of Symbolic Logic, Vol. 19, No. 1 (Mar., 1954), pp. 29-36Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2267647 .

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 19, Number 1, March 1954

A DEFINITION OF NEGATION IN EXTENDED BASIC LOGIC

FREDERIC B. FITCH

In a previous paper' it was shown that the system K of basic logic could be formulated in a simpler way owing to the fact that the proper ancestral could be defined in terms of the other concepts of that system. In the present paper analogous but more far-reaching results will be obtained for the system K' of extended basic logic.2 In particular we will show that negation and the dual of the proper ancestral, as well as the proper ancestral itself, are definable in terms of the other concepts of K'. Hence, in order to define K', we need to add only a single non-finitary rule to the rules used to define K. This rule was already among the rules originally used to define K'. It asserts that 'Aa' is in K' if (and only if) every 'b' is such that 'ab' is in K'.

We will also show that a large class of non-finitary classes and relations are represented in K', among which is K' itself, just as all finitary syntactical classes and all finitary two- and three-place syntactical relations are represented in K, one of which is the finitary syntactical class K itself. The point is that K' is adequate to handle the sort of transfinite induction that is essential in formulating K', just as K is adequate to handle the ordinary finitary mathematical induction required in defining K'.

It is easy to see that both K and K' are free from contradiction in the sense that not all U-expressions are derivable in either of these systems. The system K' is known to contain a large part of mathematical analysis.3 It is also in a sense more symmetrical than K, since it contains rules for universality, existence, conjunction, and disjunction, while K contains rules for existence, conjunction, and disjunction, but not for universality.

It can be shown that the necessary and sufficient condition for a class or a two- or three-place relation of natural numbers to be recursively enumerable is that it should be definable in K. Not only are all recursively enumerable classes and two- and three-place relations of natural numbers definable in K', but so also are all arithmetic classes and two- and three- place relations of natural numbers, and even some that are not arithmetic, namely the class of Gbdel numbers of provable formulas of K' itself. We therefore propose to apply the term "semi-arithmetic" to those classes and

Received June 9, 1953. 1 F. B. FITCH, A simplification o/ basic logic, this JOURNAL, vol. 18 (1953), pp. 317-325.

We wvill refer to this paper hereafter as SBL. 2 F. B. FITCH, An extension of basic logic, this JOURNAL, vol. 13 (1948), pp. 95-106.

This paper will be referred to as EBL. See also John R. Myhill, A linitary metalanguage for extended basic logic, ibid., vol. 17 (1952), pp. 164-178.

3 F. B. FITCH, A demonstrably consistent mathematics, this JOURNAL, vol. 15 (1950), pp. 17-24, vol. 16 (1951), pp. 121-124.

29



30 FREDERIC B. FITCH

two- and three-place relations of natural numbers that are definable in K'. A definition of "semi-arithmetic" that does not refer to K' will be presented in a subsequent paper.

We will now presuppose all the conventions and results of the paper SBL. The following rules provide the new and simpler way of formulating K'. They are a subset of the rules originally used to define K' in EBL.

Rule for '-'. '[a b]' is in K' if (and only if) 'a' and 'b' are the same U-expression.

Rule for 'e'. 'Eba' is in K' if (and only if) 'ab' is in K'. Rule for 'E". '14bac' is in K' if (and only if) 'abc' is in K'. Rule for Vo'. 'oabc' is in K' if (and only if) 'a(bc)' is in K'. Rufle for 'o". 'oabcd' is in K' if (and only if) 'a(bc)d' is in K'. Rule for 'W'. 'Wab' is in K' if (and only if) 'abb' is in K'. Rule for 'V'. '[a V b]' is in K' if (and only if) 'a' or 'b' is in K'. Rule for '&'. '[a & b]' is in K' if (and only if) 'a' and 'b' are both in K'. Rule for 'E'. 'Ea' is in K' if (and only if) there is a 'b' such that 'ab'

is in K'. Rule for 'A'. 'Aa' is in K' if (and only if) every 'b' is such that 'ab'

is in K'.

From the above rules we can derive rules for single, double, and triple abstraction just as was done in SBL for K. Also we can define the operators 'Z0', 'Z1', and 'Z2' as in SBL. For convenience we will write 'Z1' simply as 'Z'. The following rule is available and will be called "the rule for 'Z' ":

Zab " a(Za)b.

An equivalent form of this same rule is as follows:

[b E Za] " [Za a b].

We now consider a process of defining classes of U-expressions by a kind of transfinite induction, in fact the very kind of transfinite induction used in defining K' itself. Let T1, .. . , Tm be relations of classes of U-expressions to U-expressions. Relatively to these relations we can define a class C by transfinite induction by means of the following rules, taking i = 1, .. m, and assuming m finite.

Rule [Ti]. If C bears TV to 'a', then 'a' is in C. For example, in defining K' itself we can take m = 10 and choose the

relations T1, . . . , T10 as follows: Let T1 be the relation of each class of U-expressions to each U-expression

of the form '[a = a]'. Let T2 be the relation that holds from a class B of U-expressions to a

U-expression 'eba' provided that 'ab' is in B.



A DEFINITION OF NEGATION IN EXTENDED BASIC LOGIC 31

Let T3 be the relation that holds from a class B of U-expressions to a U-expression 'E'bac' provided that 'abc' is in B.

Let T4 be the relation that holds from a class B of U-expressions to a U-expression 'oabc' provided that 'a(bc)' is in B.

Let T5 be the relation that holds from a class B of U-expressions to a U-expression 'o'abcd' provided that 'a(bc)d' is in B.

Let T6 be the relation that holds from a class B of U-expressions to a U-expression 'Wab' provided that 'abb' is in B.

Let T7 be the relation that holds from a class B of U-expressions to a U-expression '[a V b]' provided that 'a' or 'b' is in B.

Let T8 be the relation that holds from a class B of U-expressions to a U-expression '[a & b]' provided that 'a' and 'b' are both in B.

Let T9 be the relation that holds from a class B of U-expressions to a U-expression 'Ea' provided that there is a 'b' such that 'ab' is in B.

Let T10 be the relation that holds from a class B of U-expressions to a U-expression 'Aa' provided that every U-expression 'b' is such that 'ab' is in B.

On the other hand, if we wish to define what might be called the class of K'-false U-expressions, we would choose T2-T6 as before, but T1 and T7-T10 would be chosen thus:

Let T1 be the relation of each class of U-expressions to each U-expression of the form '[a = b]', where 'a' is different from 'b'.

Let T7 be the relation that holds from a class B of U-expressions to a U-expression '[a V b]' provided that 'a' and 'b' are both in B.

Let T8 be the relation that holds from a class B of U-expressions to a U-expression '[a & b]' provided that 'a' or 'b' is in B.

Let T9 be the relation that holds from a class B of U-expressions to a U-expression 'Ea' provided that every U-expression 'b' is such that 'ab' is in B.

Let T1o be the relation that holds from a class B of U-expressions to a U-expression 'Aa' provided that there is a U-expression 'b' such that 'ab' is in B.

As a consequence of a theorem now to be proved, it will be shown that the class of K'-false U-expressions is represented in K'. This will then lead to a definition of negation.

Let us recall that the class represented by 'c' in a class B of U-expressions is the class of U-expressions 'a' such that 'ca' is in B. Also, the two-place relation represented by 'r' in B is the relation of a U-expression 'a' to a U-expression 'b' such that '[a r b]' is in B. The process of representing in B a relation T of classes of U-expressions to U-expressions is somewhat more complicated. We say that (a U-expression) 't' represents a relation T




(of classes of U-expressions to U-expressions) in B if for every 'c' and 'a' we have: '[c t a]' is in B if and only if the class represented by 'c' in B bears T to 'a'.

We wish to show that any class definable by transfinite induction by use of rules [Ti], where the relations Ti are chosen as relations represented in K', is itself represented in K'. Observe first if m such relations are used, we can replace them by the single relation which is the union (logical sum) of all of them. Thus we need to consider only the case where a single rule [T] is used and where T is any relation (of classes of U-expressions to U-expressions) which is represented in K'. Such a relation is represented in K' if it is the union of relations represented in K'.

THEOREM. If C is a class definable by transfinite induction by means of the rule [T], where T is a relation of classes of U-expressions to U-expressions and is represented in K' by 'I', then C is represented in K' by 'Zt'.

In order to establish this theorem let us define D as the class represented by 'Zt' in K' and then show that D has the same members as the class C which is defined by means of rule [T]. If D bears T to 'a', then, since 't' represents T in K', we infer that '[Zt t a]' is in K'. From the rule for 'Z' it then follows that 'Zta' (or, equivalently, '[a ? Zt]') is in K' and hence that 'a' is in D. Thus anything that can be shown to be in C by use of the rule [T] can also in a parallel way be shown to be in D. In other words, D itself satisfies the rule [T]. Conversely, if something can be shown to be in D, it can by means of the rule [T] be shown to be in C. For suppose that 'a' can be shown to be in D. This means that 'Zta' can be shown to be in K'. But in order to show that 'Zta' is in K', it is first necessary to show that '[Zt I a]' is in K' (just as '[x(... .x... )]a' cannot be shown to be in K' without first showing that '(. .. a. . .)' is in K'). But to show that '[Zt t a]' is in K' is in effect equivalent to showing that D bears T to 'a'. Thus if 'a' can be shown to be in D, it must be because, in effect, D can be shown to bear T to 'a'. Consequently, if 'a' can be shown to be in D, then 'a' can in a parallel way be shown to be in C. Hence C and D must have the same members.

From the above theorem it follows that the class of K'-false U-expressions is represented in K' by 'Zt', provided that 't' is chosen in such a way that it represents in K' the union of the relations T1, . . . , T10 involved in the definition of the class of K'-false U-expressions. In order to find such a 't' we first find U-expressions 't1', .. ., '110' which respectively represent in K' these relations T1, . .. , T10. This is easily done. For example, 't,', 't2', 't9', and 't10' can, by use of the rule of double abstraction, be chosen in such a way that we have:4

4 The inequality symbol and the existence quantifier are to be understood ais define cdi in SBL. The universal quantifier is to be understood as defined in the same way as the existence quantifier, but using 'A' in place of 'W'.




t1be (3x, y)[[e [x yfl & [x t y] & [b - b]], t2be (3x, y)[[e- Eyx] & b(xy)], t9be (3x)[[e Ex] & (y)(b(xy))], tlobe +-+ (3x)[[e Ax] & (3y)(b(xy))].

We then choose 'I' so that

the +-+ [t1be V t2be V t3be V ... V t10be].

If, for such a 'I', we treat the negation symbol, "',, as an abbreviation for 'Zt', then the class of K'-false U-expressions is represented in K' by ', and the following rules clearly become derivable: Negative rule for '=='. '.'[a = b]' is in K' if (and only if) 'a' and 'b'

are different U-expressions. Negative rule for 'e'. '.(Eba)' is in K' if (and only if) '.(ab)' is in K'. Negative rule for 'e'. '(ebac)' is in K' if (and only if) '-(abc)' is

in K'. Negative rule for 'o'. '.(oabc)' is in K' if (and only if) 't-...(a(bc))'

is in K'. Negative rule for 'o". '-(o3abcd)' is in K' if (and only if) '-(a(bc)d)' is

in K'. Negative rule for 'W'. '.(Wab)' is in K' if (and only if) '-(abb)' is

in K'. Negative rule for 'V'. ','a V b]' is in K' if (and only if) ',a' and 'fib'

are both in K'. Negative rule for '&'. [a & b]' is in K' if (and only if) ',a' or 'nb'

is in K'. Negative rule for 'E'. ',(Ea)' is in K' if (and only if) every 'b' is such

that '-(ab)' is in K'. Negative rule for 'A'. '"(Aa)' is in K' if (and only if), there is a 'Y

such that '.(ab)' is in K'. From the above rules we can derive negative rules for single, double, and

triple abstraction and negative rules for 'ZO', 'Z', and 'Z2'. The negative rule for 'Z', for example, would be:

,(Zab) "(a'(Za)b), or, [b E Za] " [Za a b].

We next wish to define '*'. The method used in SBL apparently fails to provide the required negative rule for '*'. Accordingly we will use a somewhat different method. Some preliminary results must first be presented.

ILet 'L' be

[L (3I, v, zv) [[y = uvwl 6'[FK - _ V ( [ z & xx()tvz)]],]'.1

Thlien it is easy to show that

LL(/ab) - [[b = a] V (3z)[[b -- fz] & LL(jaz)]j.




Let 'N' be such that Niab e-+ LL (lab).

Then we get

Nlab [[b = a] V (3z)[[b -z] & N/az]],

where the left side can be shown to be in K' only if the right side has first been shown to be in K'. By use of the negative rules we also get

,-'(N/ab) en [[b + a] & (z)[[b + /z] V (N/az)]],

where again the left side can be shown to be in K' only if the right side has first been shown to be in K'. From these results we can obtain the following two rules:

Rule for 'N'. 'Nlab' is in K' if (and only if) 'b' is one of 'a', 'la', '/(/a)', f/ (/ (la))', . . . Negative rule or 'N'. '-(N/ab)' is in K' if (and only if) 'b' is no one of

'a', (lfa), ' f (a) ', '/(/(/a)) ', It is easy to find U-expressions 'D1' and 'D2' such that

D1/(ab) "-> lab,

D2/g(ab) +--> (3z)[/(az) & g(zb)].

Let '(/1)' be a temporary abbreviation for 'D1/', and let '(/ui+1)' be a temporary abbreviation for 'D2/(/n)', where n 1, 2, 3, . Then we must have

(/n)(ab) - (3x1, x.-. * ) [lax, & /XlX2 & . &xn1b],

and also,

t__((1n)(ab)) "(X1) XnJl[--(f1xj) V --(flXl2) V ..V --(/Xn-jb)].

Now from the rule for 'N' we see that 'N(D2/)(Dl/)e' is in K' if and only if e' is one of '(/1), '(2)', '(3)' ... Also, from the negative rule for 'N' we see that '/-.(N(D2/)(Dl/)e)' is in K' if and only if 'e' is no one of these latter expressions. Let us now define '*' as

'[FV z (3 w) [N (D 2X) (D1x)zv wyz)`.

Tlien we have */Iab *-> (3zw)[N(D21))(Dl)zv &wze(ab)],

-(*/ab) (w) [-(N(D2l) (Djl)7) V '-.(we(ab))].

Hence, '*lab' is in K' if and only if at least one of '(11)(ab)', '(12)(ab)', '(f/)(ab)', ..., is in K'. Also, ',(*lab)' is in K' if and only if each of ',((/1)(ab))', '_((l2) (ab))', , .(.(.),'is in K'. Consequently the following two important rules are derivable:




Rule for '*'. '*lab' is in K' if (and only if) for some n > 1, there is a finite sequence 'cl', 'c2' . . ., 'ca' such that 'cl' is 'a', and 'c,' is 'b', and'/c1c2',

lC2C3' ... , '/lc,1c' are all in K'. Negative rule for '*'. ',(*lab)' is in K' if (and only if) for every finite

n > 1, every sequence 'cl', 'c2', ..., 'ca', where 'cl' is 'a' and 'c)' is 'b', is such that at least one of C'(/cc2)', 'U(/C2c3)'. -.. I '.(/1clllc)' is in K'.

The above two rules are essentially the same as the corresponding two rules given on page 97 of EBL as rules [*] and [-*]. The dual of the proper ancestral can be defined analogously to the proper ancestral itself, and its two rules can be derived in a similar way. The present definition of '," thus gives all the rules used as defining rules for K' in EBL, except the rule of double negation. But this deficiency can be remedied by modifying the definition of negation in the following way. Instead of treating '--' as an abbreviation for 'ZI', as was previously done, we now treat it as an abbreviation for 'Z(Yt)', using the same '1' as before, and choosing 'Y' so that

Ytda - [Zia V (3x)[x-& [a - dxl]].

Then Z(YI)a - YI(Z(Yt))a

" [Zia V (3x)[x & [a - Z(Yt)xl]].

We can then derive all the previously stated negative rules and also the following rule of double negation:

Negative rule for ' '-(-a)' is in K' if (and only if) 'a' is in K'. This completes the demonstration that all the rules previously used in

defining K' are derivable if the new and simpler definition of K' is used. The method used to define negation in K' can of course also be used to

define in K' an expression representing K' in itself, or representing in K' any system that is formulated by the same sort of procedures used in formulating K'. (But obviously the simplest way to represent K' within itself is to choose the representing expression as '[]x]', that is, 'W&'.) At least one such system, however, is not "completely represented" in K' in the sense explained in EBL, since it was shown in EBL that K' is not completely represented in itself. There are many unsettled questions in this connection. For example, if a class and its complement are both represented in K', is that class completely represented in K'?

The method of defining classes by rules of the form [T] can with slight modifications be used to define two-place and three-place relations which are represented in K' if the corresponding relation T is so represented. For example, a three-place relation R is definable in this way if a suitable rule [Tj can be found for defining the class F of all U-expressions 'abc' such that R relates 'a', 'b' and 'c' in that order.




By adding the rule for 'A' respectively to the systems K* and Kt of SBL, we get systems K*' and Kt' which are to K' very much as K* and Kt are to K. In particular, K*' is a minimum basic non-finitary system in a sense analogous to the sense in which K* is a minimum basic finitary system.

YALE UNIVERSITY



a definition of negation in extended basic logic

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