w.v. quine, encyclopedia american a, selected logic papers, symbolic logic, ii

7/31/2019 W.v. Quine, Encyclopedia American A, Selected Logic Papers, Symbolic Logic, II

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36 Selec ted Logic Paperstion, an d foreshadow to some degree the projected volume ongeometry; and a continuation is outlined in Whi tehead 's laterwritings61 und e r th e he ad of "extensive abstraction." Other con-structions in the 1906 paper go far outside geometry; this w as th ebeginning of a quest for the broadest , most basic concepts andprincipies of nature , and in the decades since Principia th e quesths issued in a metaphysics."1916; 1919; 1920; 1929.

II

Logic , SymbolicNotations reminiscent of lgebra were used sporadically in de-ductive or formal logic f rom Gottfried von Leibniz onward, an dincreasingly in the 19th century (George Boole, Augustus De-Morgan, William Stanley Jevons, Charles Sanders Peirce). GottlobFrege, in 1879, supplanted the algebraic notation by anothe r wh ichbetter accentuated the structural traits most ge rmane to deduction.Flourishing like other happily f ormulated branches of mathematics,deductive logic soon s exceeded i ts earlier pow ers as to invite re-christening: whence "symbolic" or "mathematical" logic.If in a sentence we put dum my letters fo r ali portions other thanth e logical part icles 'or', 'and', 'not', 'if, 'every', 'some', and thelike, w e g e t roughly what may b e called a logical form. A form iscalled valid if ali sentences having that form ar e t rue. The logicalruths, finally, are the sentences valid in fo rm; and the concern ofdeductive logic, or symbolic logic, is systematic recognition of logi-cal truths.When a logical truth hs the form oi a conditional ('If . . .then . . .'), the one clause is said to imply the other logically.Implication underlies th e practical use of Deductive logic, namely,deduction: th e infe rr ing of sentences, on any subject, from an ysupposed truths which logically imply^them.The above description is inexact, mainly because of the imperf ec tinventory of the logical particles. A more adequate accounting ofthese particles will emerge as we get on with the sub stantive theoryof symbolic logic in the following pages.This article was w ritten in 1954 and is reprinted from Encyclopedia Americana,1957 and later editions, by permission. The Encyclopedia reserves ali righta toreproduction of this article elsewhere.


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38 Se l ec t ed Logic Papers1. Truth-Function Logic, or Propositonal Calculus. The conjunc-tion of any sentence s is a sentencecompounded of them by juxtaposi-

tion, let us say, and construed as true i f and only if the componentsentences are al i true. The alternation of any sentences, fo rmed by aconnective 'V, is false if and only if the components are ali false.The negation of a sentence, formed by writing '-' before or above, istrue if and only if the sentence itself is false. Thus juxtaposition,' V ', and '-' may be read 'and', 'or', and 'not', though in abstrac-t ion f rom such verbal usage as may diverge from the rigid truthcondi ions just no w stated.

Thus, let 'p', 'q', and V be dummy sentences. Then 'pq V pj Vpf' will be true if and only if at least one of these three combinationsis realized: 'p' an d 'q' both true, or both false, or 'p' true an d Vfalse.A compound sentenceis said to be a truth functon of its compo-nent sentences if the truth value (truth or falsity) of the compoundis determined by the truth values of the components; that is, if nosubstitution of truths for the component true sentences and false-hoods fo r the false ones will alter the truth value of the compound.Obviously any compound is a truth functon of its components ifbuilt up of them by conjunction, alternation, and negation. It iseasily shown also, conversely, that conjunction, alternation, andnegation provide a complete notation for the truth functions.

Alternation ca n even be dropped, since 'p V q' can be rendered'-(p?)'. But the threefold notation is convenient. It enables us torender an y truth funct ion of given components 'p', 'q', etc., in theperspicuous alternational normal form; that is, as an alternation ofconjunctions of elements from among 'p', 'p', 'q', 'q', etc. Transfor-mation into that form is accomplished by distribution according tothree laws: '-(s V f V . . . )' becomes 'si . . .'; '-(st . . . )' be-comes 's V V . . .'; and V (s V t V . . . )' becomes 'rs V ri V. . .'. (Terminological note: we speak of conjunction not only ofmany components but also of one, thus counting anything a con-junction of itself, and similarly fo r alternation; s 'p', 'pq', 'f V q' ,etc., count as in alternational normal form.)Everything said in the foregoing paragraph remains true whenw e systematically switch the roles of alternation andconjunction.Thus any truth funct ion can be put also into a conjunctonal normalfo rm. This so-called duali ty between alternation and conjunctionrests on the f ac t that their truth conditions are alike except for asystematic interchange of 'true' and 'false'.

L O Q I C . S Y M B O L I C 39The compounds 'p V q' (or equivalently '-(pq)') and 'pq V f$are commonly rewritten 'p 3q' and 'p ss q' respectively, andcalled the (material) conditional and biconditional, and read 'if p

then q' and 'p if and only if q' . Though they mirror only imperfectlythe vague idioms 'if-then' and 'if and only if', they suffice for muchof the business of those idioms. Eked out by "quantification" (seebelow), they suffice for more of it stillindeed for ali of it but thequestionable business of the subjunctive or contrary-to-fact condi-tional. The importance of truth-function theory is due largely to theconnective Oand its intimate connection with logical implication,namely this:one sentence implies another if and only if the materialconditional formed f rom the two sentences is logically true. Theconnective '=' is similarly related to logical equivalence, or mutualimplication.

In order to avoid excessivo parentheses, dots are used to markmajor breaks in formulas. A dot, when used instead of mere juxta-position to express conjunction, marks a grater break than ' V','D, or s;any of these connectives with a dot added marks agreater break still; a double dot marks a grater break still; and son. Thus 'p V. p V q.pq V r p' stands for '{p V [ (p V q)(P? V r)]} D p'.

We can evaluate (that is, determine the truth value of) any truthfunction of 'p', 'q', etc., for any given assignment of truth values to'p', y etc.; by applying the truth conditions forconjunction, alter-nation, and negation step by step. Validity of a formula of the logicof truth functions (for example, 'pq V pf V p> V ps V f r V fS')is t her efor e decidable by tabulating ali possible assignments oftruth values to letters of the formula and evaluating the formulafor each. Such is the method of truth abks (which in practice admitsof certain shortcuts).

Implication between formulas of truth-function logic is likewisedecidable by truth tables; for one formula implies another if thecorresponding conditional is valid. Equivalence is then decidable inturn by two implication tests. Or, more simply, we may test twoformulas fo r equivalence by constructing a truth table for each andthen comparing the truth tables to see if they are the same.

Simple examples of implication in the logic of truth functions arethese: 'pq' implies 'p'; 'p' implies 'p V q'; 'p. p D q' implies 'q'.Each, characteristically, repeats 'p' and s could quickly engenderfallacy if a sentence substituted for 'p' were capable of being true atsome points and false at others within the same train of logical


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40 Seleced Logic Papers L O G I C , S T M B O L I C 41argument . To avoid this fal lacy of equivocation we do not need to f ixali ambiguities in the sentences to which logic is applied, nor alire fe rences of pronouns an d demonstratives; but we do need to fixsuch of them as might otherwise vary by force of context within th espace of the argument.Truth- func t ion logic was pursued by the Stoics and by PetrusHispanus and William of Ockham. Reappearing as a variant of thelgebra of classes (next topic), it was perfected by Peirce, Frege,Ernst Schrder, and Emil L. Post.. lgebra of Monadic Pred ica te s or Classes. General terms orpredicates are roughly nouns, adjectives, and verbs. This gram-matical trichotomy is indifferent to logic; a more germane distinc-tion is that between absolute general terms or monadic predicates(for example, 'man', 'red', 'thinks') and relative general terms orpolyadic predicates (for example, 'uncle of, 'greater than', 'takes').Sentences ar e true or false; monadic predicates ar e true or false ofobjects; dyadic predicates ar e true or false of pairs of objects; an ds on . Accordingly th e truth funct ions, which apply to sentences,have analoguescalled Boolean funct ionsapplicable to predi-cates: a conjunction ("logical product") of predicates is true ofwh a t ev e r the predicates are jointly true of ; an alternation ("logicalsum") is true of what at least one of the predicates is true of ; and anegation is true of what th e predicate is false of .In the lgebra of monadic predicates, also called class lgebra, w ebuild Boolean functions of dummy predicates 'F', 'G', etc. Also tw ospecial predicates emerge, 'V and 'A' , true of everything andnothing. Joining any such expessions by ' = ', we get Boolean equa-tons. An appropriate reading of 'F = (?' in general is 'Ali F ar e Gan d vice versa'; hence in particular 'FG = A' amounts to 'No F ar eG' , an d 'FG = A' to 'Ali F ar e G' . 'FF = A' an d 'F V F = V a r evalid, or true fo r ali interpretations of 'F'. A ny Boolean equationhs an equivalent with 'V as one side, and another with 'A ' as oneside; for 'F = A' is equivalent to 'F = V, 'F = V to_'F = A' ,and, in general, 'F = G' to 'FG V FG = V and to 'FG V FG =A'.This lgebra was developed mainly by Boole, Jevons, andSchrder. Its formulas (valid and otherwise) comprise the Booleanequations an d ali truth functions of them. Valid examples include:(1) FQ* A.G# = AO-FH 7 * Aand the like, answering to syllogisms in traditional logic.

What things 'V is to be true of, and what things 'F' is to be trueof, given an interpretation of 'F', depend on our choice of "universeof discourse" (DeMorgan). This choice may convniently be variedf rom application to application; s formulas are counted valid onlyif true under ali interpretations of 'F', 'G', etc., in al i non-emptyuniverses. The empty universe is profitably excepted because someformulas, for example 'V 5^' A', fail for i t which hold generallyelsewhere. The question whether a formula also holds fo r the emptyuniverse is easily settled, whe n desired, by a separate test; for al iBoolean equations hold true for the empty universe, and accord-ingly any truth function of them can be tested by "evaluation" (seeabove).

Validity is of interest mainly as a key to implication, which isvalidity of the conditional. Now in simple cases implications areeasily checked directly, by diagramming classes as overlappingcircles (John Venn). To explore th e consequences of given Booleanequations and inequalities we shade regions of the diagram whichthe equations declare empty, and flag regions which the inequalitiesdeclare occupied, an d observe th e effects.

But the prime desideratum is a general decision procedure fo r thispart of logic, such as the truth table provides for truth-functionlogic; i.e., a mechanical method of testing an y truth function ofBoolean equations for validity. The earliest such procedure wasestablished by Leopold Lwenheim; others, easier, ha v appearedsince. One is as follows.Let us suppose ali equations (and inequalities) standardized s

as to have 'A ' as right-hand member, and no 'A ' nor 'V in the lef tmember. (Occurrencesof 'A' and 'Von the left can be got rid of bysubstituting 'FF' an d 'F V F', or by a more efficient method whichneed not detain us.) Now any alternation of (one or more) such in-equalities, e.g.:(2 ) -(FG) j A..V.GH 7 * A . V . F f f ^A,can be tested for validity as follows: delete '^A' everywhere, andtest th e result ('-(FG) V GH V F H, in the case of (2)) fo r validityby truth table (as if 'F', 'G', etc., were sentence letters).

The procedure can be extended to any alternation of formulaswhereof one is an equation and the rest (ifany) are inequalities, asfollows: weaken the alternation by changing its one equation to aninequality and negating the lef t member thereof; then test the thusweakened alternation as in the preceding paragraph. For example,


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42 Selec ted Logic Papers'FG = A .V. GH 9 * A .V.FH ^ A' (which is the syllogism (1),converted to normalform) weakens to (2). Such weakening s by nomeans an equivalence transiormation; yet i t does leave non-validformulas non-valid and valid ones valid, as long as we adhere toalternations with only on e amrmat ive equation. (This is not evi-dent, but can be proved.)Th e procedure can be extended in turn to any alternation ofequations and/or inequalities; for an alternation which includesseveral amrmative equationsis valid if and only'if, by droppingalibut a certain one of i ts aff irmative equations, i t can be got down toan alternation valid according to the preceding paragraph. (Thisagain is not evident, but can be proved.)Finally we can test an y truth function whatever of Boolean equa-tions, as follows. We put the whole into conjunctional normal form,handling its equations as onewould sentence letters. The result is aconjunction of alternations as of the preceding paragraph; and thewhole is valid if and only if each of those alternations is valid.

S. Quantification. Conjunction, alternation, and negation havetheir analogues fo r polyadic predicates, as for monadic predicates.B ut there ar e also fur the r operations, applicable to polyadic predi-cates, which have no analogues fo r sentences or monadic predicates.On e is converse, which, applied to 'greater than' or 'parent', gives'less than' or 'offspring'. Another is relative product , lustrated by'friend of fa the r of . Another is image, which combines dyadic an dmonadic predicates as in 'friends of musicians'. Th e lgebra ofdyadic predicates or relations (DeMorgan, Peirce, Schrder) , whichtreats of these matters, is far more complexan d less intuitive thanth e lgebra of monadic predicates or classes. On the other hand th elogical inferences fo r which this extended lgebra provides, an dfur the r ones fo r which it does not, prove easily manageable under amore analytical approach which departs from th e pattern of anlgebra of predicates; namely, th e logic of quantification, foundedby Frege.Th e universal quantifier '(x)' may be read 'Each object x is suchthat', and the exis tent ial quantifier '(3x)' may be read 'At least on eobject x is such that'. These are prefixed to open sentences such as'x is red ' which, of themselves, are neither true no r false becauseofth e free 'x'. Th e resulting quantifications in this instance, '(x)(x isred)' and ' (3z)(x is red)', are closed sentences and are respectivelyfalse an d true.

L O G I C , S Y M B O L I C 43Writing 'Fx' to indicate application of the predjcate 'F' to 'x', w ecan express 'Ali F ar e G' and 'Some F ar e G' as '(x)(Fx DOx)' and'(3x)(Fx . Ox) ' , an d 'No F ar e G' indifferently a s ' ( x ) ( F x D -Gx)'or '-(3x)(Fx . Gx)'. Typical purposes of Boolean equations can thusbe served alternatively by quantification and truth functions. In-deed an y Boolean equation can be translated into these terms; firstwe rephrase it with 'V as its right side, and then wetranslate it as'(*)( . . . )' where '. . .' is got from the left side by inserting 'x 'after each predicate letter.But the great value of quantification theory resides in the factthat it is adequate equally to polyadic matters. Here we write predi-cate letters followed by multiple variables: 'Fxy', 'Gxyz'. The effectsof the above notions of converse, relative product, an d image arethen go t by 'Fyx', '(3y)(Fxy . Gyz)', and '&y)(Fzy .Gy)'.The use of variables other than 'x ' in quantifiers changes thesense of the quantifiers in no way, but serves merely to preservecross-references, as in '(3x)(y)Fxy'; this is the quantification by'(3z)' of the open sentence '(y)Fxy'.The valid formula '(3x).(y)Fxy D (y)(3x)Fxy' illustrates the ex-tended coverage which quantification theory affords as contrasted

with th e lgebra of monadic predicates or , afortiori, th e traditionaltheory of the syllogism.Th e formulas of quantification theory are built up from atomicformulas 'p', 'q', 'Fx', 'Fy', 'Gxy', 'Hyx', 'Jxyz', etc., by truth func-tions an d quantification. The convention is that the atomic for-mulas represent any sentences, subject to these conditions: (1) Ifthe atomic formula occurs under a quantifier whose variable itlacks, then th e sentence which it represents must lack f ree occur-rences of that variable. (2) If two atomic formulas are alike exceptfor variables (e.g., 'Fx' and 'Fy'), then the represented sentencesmust be similarly related.Interpretation of a formula consista in choosing a universo asrange of values of 'x ', 'y', etc., choosing truth values fo r 'p', 'q', etc.,choosing specific objects of the universe for any free variables, an ddeciding what objects (o r pairs, etc.) th e predicates 'F', 'G', etc.,are to be true of. A formula is valid if true under ali interpretationswith non-empty universes. As before, the case of the empty uni-verse is easily handled by a separate test; for ali universal quanti-fications are there true, and al existential ones false.Implication, as usual, is validity of the conditional. In particular'Fyy' implies '$x)Fxy'. This example of implication, and closely


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44 Se l ec t ed Logic Papers L O G I C , S Y M B O L I C 45similar ones, will now be given a special name. The existentialquantif ication of any formula , with respect to a variable a, will becalled an existential consequence of any formula which is like ex-cept fo r having free occurrencesof some variable /3 wherever hsfree occurrences of a,By a surface occurrence of a formula < t> in a formula ^ let us under-stand an occurrer ce wh ich is overlaid by no quantifier or negationor O or 's-'; nothing but conjunction and alternation. Now itis readily seen that th e following rule of inference leads f rom formu-las always to formulas which they imply.

RULE I: Supplan a surface occurrence of < j> in $ by an existentialconsequence o f < j > .Example: two steps by Rule I lead f rom '-Fwz V Fwz' to '(3y)-FwyV ( lx)Fxz'.The next rule of inference does not lead always to impliedformulas, but it does lead f rom valid formulas to none but validformulas.

RULE II : If a is a variable whose free occurrences in ^ li e whollywithin a surface occurrence of < t > , then inseri a universal quantifiers as to quantify < with respect to a. In s doing you may also changea to a ne w let ter $, provided that jS is foreign to < j > .Example: two steps by Rule II lead f rom the result of the pre-ceding example to '(x)(3y)-Fxy V (y)(3x)Fxy'.The next two rules obviously lead f rom formulas to equivalentformulas.

' RULE III: Supplan t an y clause ai will by any which is equivalentto it by t ruth tables.RULE IV : Change '(3)-', '(*)-', '&/)-', etc., at wil l to '-(x)','-(3x)', '-(y)', etc ., respecively .

Example: two steps by Rule IV lead f rom th e result of the pre-ceding example to '-(3x)(y)Fxy V (y)(3x)Fxy', or '(3x)(y)Fxy 3(y)(?x)Fxy'.As the above chain of examples illustrates, Rules I-IV enable us,starting w ith a form ula wh ich is valid by truth table ('-Fwz V Fwz'in the example), to generate fur the r valid formulas of quantifica-tion theory. This technique is in fact complete (Kurt Gdel, JacquesHerbrand) , that is , capable of yielding an y valid fo rmula.It is convenient in practice, though strictly unnecessary, to

invoke supplementary rules, based, for example, on the implicationof 'Fy' by '(x)Fx', or the equivalence of '(x)(Fx. Gx)' to '(x)Fx .(x)Gx' or of 'Qx)(Fx V Gx)' to '(3x)Fx V ( lx)Gx'.Despite its completeness, the described proof procedure is not adecision procedure (see above). For, e ven if we fail to find the proofof a formula, we may not know whether the formula is neverthelessvalid, its proof having merely eluded us. In truth-function logicand again in the lgebra of monadic predicates we sa w decisionprocedures; for quantification theory, however, none is possible(Alonzo Church).

4. Identi ty. The sign '=' wasused in earlier pages to form truesentences f rom coextensive predicates. But it hs its primary userather be tween variables 'x ', 'y', etc., as a dyadic predicate in itsow n right, expressing identity. The theory of identity can besummed up in the axiom '(x)(x = x) ' and the axiom schema'(x)(y)(x = y . Fx O- Fy)', wherein 'Fx' and 'Fy' represent anysentences which are alike except that the one hs free 'x' at somepoints whe r e the other hs free 'y'.Gdel showed that every truth in the notation of identitytheory is obtainable f rom this basis b y quantification theory. Fo rexample, one case of the axiom schema is '(x)(y)(x = y .x=x O. y x)', which, with '(x)(x = x) ', yields '(x)(y)(x = y O.y = x)'.Identity, added to the truth functions and quantifiers, enablesus to deal with the idioms 'only x', 'everything else', and the like.From 'Someone on the team admires everyone else on the team'and 'Some fielder on the team is admired by no one' we can argueby means of quantification theory plus identity and its axioms,but not by means of quantification theory alone, that some fielderon the team admires evryone else on the team.The adding of identity provides also for a rudimentary treatmentof number: we can express 'There ar e exactly n objects x such thatFx', symbolically '(3 nx)Fx' , fo r each fixed n. For, ' ( 3 < , x ) F x ' can berendered '~(3x)Fx', an d '(3 nx)Fx' for each succeeding n can be'rendered '(3x)[Fx . (3 n-iy)(Fy . x ^ y)]' (Frege).Th e adding of identity enables us also to introduce an operatorof singular description '(tx)' (Frege, Giuseppe Peano) in such a wayas to serve th e useful purposes of the words 'the object x such that'.The trick (essentially Bertrand Russell's) is to explain '(ix)Fx'not outright, but in any atomic context. Thinking of 'G(ix)Fx' as


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46 Se l ec t ed Logic Papersany such context of '( ix)Fx', w e expiam 'G(ix)Fx' as short fo r'(3y)[Fy . Gy , (x)(Fx D x = y}]'; that is, some F is Gand nothingbut it is F. This device enables us to handle singular terms withinthe f r a mewo r k of pur quantification theory with identity. Con-sider, for example, the function sign or operator '+', which pro-duces compound singular terms of the form 'y + z'. Its purposesca n be served by an ordinary triadic predicate '2', where '2xyz'means that x is the sum of y and z; for, 'y + z' can then be taken as'( ix)2xyz', subject to the general contextual definition of '( ix)Fx'.

5. Set Theory. In '=' we have a logical predicate. There is an-other, th e predicateV of class membership, which is more powerful:s much s that it may with some justice be viewed rather asmathematical, in a sense exclusiveof logic. Once V is assumed, ' = 'is dispensable; for 'x - y' can be paraphrased as ' ( z ) (x ez.D.y e z}'. Much else also becomes expressible, for which '= , quanti-fiers, and truth funct ions were inadequate. For example, we canno w express 'ancestor' in terms of 'parent'. For, one's ancestors (iffor simplicity w e reckon oneself among them) comprise th e com-mon members of ali classes which contain oneself and aliparentsofmembers (Frege); an d 'x is ancestor of y' can accordingly berendered '(z)[y e z (u)(w)(u is parent of w . w z O- u e z) Ox t z]'.The logic of V is set theory, or class theory, in a more genuinesense than was the lgebra of monadic predicates. In the latterthere is no call to posit a realm of classes. 'F', 'G', etc., stand inplace of predicates, but a pedicate, like an open sentence, issimply an expression which is true of many or one or no objects.With the advent of V, on the other hand, classes are demanded asactual members of the universe of quantification; compare the 'z'of th e last example.At this stage t her efor e there is a place for class names, as againstmere predicates. Such names are formed by the notation of c lassbstraction: x Fx is the class of those objects x such that Fx . Actuallythis notation can be introduced in terms ofprior notations, as shortfor '(iy)(x)(x f y .= Fx)' . In particular 'x(x = x) ' an d 'x(x ^ x )'are abbreviated 'V and 'A'; but this use of these signs as namesofclasses is not to be confused with our earlier use of them as predi-cates. Similarly the Boolean funct ions can be explained now forclasses: xy as (z a x . z ey), x V y as (z e x . V. z ey), and x as-(z x) . Th e lgebra of monadic predicates can of course be read

L O G I C , S Y M B O L I C 47from the start with 'F', 'G', etc., as class variables and 'F = G' asordinary class identity; but to do s is a graiuitous positing ofentities before necessity.Rela t ions are understood as classes of ordered pairs (or triples,etc.). Now the one thing demanded of a concept of ordered pair,x;y, is that x;y = z; w O.x = z . y = w. This is demonstrablyfulfilled by any of the various artificial definitions of 'x;y' within thetheory of classes (Norbert Wiener). For example, having defined{x } as (z = x) an d {x , y }^ as \x ] V {y}, we can take x;y as {{a;},{x , y } } . Thereupon relational bstraction, 'xyFxy', becomes de-finableas '(3z)(3y)(z = x;y . Fxy)' . The theory of relations is thusobtained within set theory. Analogues, in particular, of the notonsof the lgebra of dyadic predicates are forthcoming: the converseofa relation z is xy(y;xt z), the relativo product of u into w is xy$z)(x;z c u . z;y e w ), and the image of w by z is (5y)(x;y e z . y t w).

Since the algebras of predicates cover less than quantificationtheory itself, their reproducibility in set theory is no motive for settheory. The gain in power afforded by V and classes is seen ratherin the ancestral construction (above), and again in connection withnumber , to which we now turn.

In defining '(3 nx)Fx' for each fixed n we made no provision fornumbers as values of variables of quantfication. In set theory,however, we can sprovide, construing each number as the classofthose classes having that number of members (Frege). Thus O isdefinable as {A}, and in general w + l, or 8w, is de&jable as$(lx)(x e y . y -{x\ w ) . Thereupon, following th e plan ojNthe defi-nition of ancestor, we can explain 'Num x ' ('x is a number') as shortfo r '(z)[0 te.(w)(wtz -D. S w z ) O. x e z] ' (Frege).

Now we have the means of saying, for example, not only that aclass x hs five members, (3sz)( t x) or a ; e 5, but also tha^z hsjust as many members as y; viz., (3z) (Numz.xez.yee). We cango farther: we can express the whole of number theory. For ex-ample, x + y is definable as comprising those classes z such thatpart of z belongs to x (i.e., hs x members) and the rest to y; sym-bolically, (3to)(zu> c x . zw y). Th e product x -y is definable, morecomplexly, by exploiting the fact that if a class z hs x mutuallyexclusive classes as members, and each of these hs y members,then there ar e x -y members of members of z.

Numbers, as thus far considered, are the sizes of finite classes,that is, the positive integers and 0. But Georg Cantor's theory ofinfinite class sizes, or infinite numbers, can likewise be expressed


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48 Selec ted Logic Paperswith in ou r present terms; a main step towards ij t is the easily de-fined notion of a one-one relation.Finite numbers of othe r sorts than O, l, 2, . . . are manageableas well. Somew hat artificially we can identify ratios %, %, etc.,with the pairs 1;2, 2;3, etc., and then redefine sum, product, andother relevant no tions appropriately for application to ratios sconstrued. Irrat ional numb ers prove to b e satisfactorily identifiablewith certain infinite classes of ratios (Richard Dedekind), andfinally negative and imaginary n umbers can be accommodated b y

' furthe r recourse to the device of ordered pair. Mathematical func-t ons can be identified with certain relations, namely, the relationsof values to arguments of the functions ordinarily so-called; fo rexample, the function "square of," symbolically X x ( z 2 ) , is definableas yx(y = x 2). By such methods, set forth in detail by Frege,Peano, Alfred North Whitehead, Bertrand Russell, an d others, se ttheory is shown to embrace classical mathematics in a very generalsense of the term. The truth functions an d quantifiers and V thusemerge as, in theory , a comprehensive notation for mathematics.Th e question of suitable axioms for 'e' is, in effect, therefore, aquestion of suitable axioms fo r mathematics. To begin with thereis the axiom of extensionality: (w)(w t x .= . w t y) . x t z O.y ** This is clearly wanted where x an d y are classes; and it can beadopted without restrict ion if we identify each non-class x , arti-ficially but conveniently, with {x}. Now what are wanted by wayof fur the r axioms are those of class existence: axioms of the forra'xFx exists', that is , '(3y)(x)(x y .= Fx)'. But actually w e can dis-prove on e such sentence, thus: '-[y e y . = -(y y ) ] ' is valid byt ruth table, an d f rom i t by Rule I and R ule II we get '(y)(3x)-[ x e y . ss -(x t x}]', an d s , by Rule IV, ' - (3y)(x)[x e y .= -( x t x)]'(RusselPs paradox). Infinitely many such counter-examples areknown. Nor can we somehow banish just such refutable cases andkeep '( 3 y) (x ) (x e y .= F x * ) ' for the rest; for various of the remain-ing cases, not refutable individually, are mutually inconsistent.

Varied systematic proposals have bee n made, since 1903 (Frege),in an effort to encompass an in some sense optimum set of consist-en t cases of '(3y)(x)(x e y .= = Fx)'. One system, RusselPs theory oftypes, stratifies the uni verse into individuais, classes of individuais,classes of such classes, and s on, and then, appropriating a dis-tinctive style of variable to each such type, re jects as meaninglessan y formula containing V otherwise than between variablesappropriate to consecutive ascending types. The cases of '(3j/)(x)

L O G I C , S Y M B O L I C

)' which survive as meaningful ar e consistent. Butthe result ing theory is unwieldy in certain rspects; moreover itproves to require supplementation with an axiom of the infinitudeof individuais, if we are to preserve the law 'x + l = y + l O.x = y' of number theory.Ernst Zermelo's method was to assume ali cases of the form'(3y)(x)(x ty .ss.xtz , Fx)' (Aussonderungsaxiom) plus a certainassortment of further cases by way of providing classes z for the

Aussonderungsaxiom to operate on. This method hs equallyserious drawbacks.John von NeumamVs method was to declare some classes in-capable of being members. Thereupon '(3y)(x)(x t y .= Fx)' ca nbe assumed in general with 'x ' restricted to elements, i.e., classesof the kind capable of membershp. (He did not assume this much,but we may.) It remains, in such a theory, to instituto conditionsof elementhood. Von Neumann modeled hi s conditions of element-hood on Zermelo's conditions of class existence; Willard V. Quinean d Ho Wang modeled theirs rather on certain traits of the theoryof types.Choice among such alternativo foundat ions of set theory hingeson relative naturalness, elegance, convenience, power, and likeli-hood of consistency. Consistency proofs are not absolute, sincethey assume the consistency of the theory in which they are con-ducted; but a theory is occasionally bolstered by a consistencyproof relative to a less suspect theory.Th e clear optimum in set theory is not at hand, and, in onesense, never can be: for Kurt Gdel h s proved that no theoryadequate to expressing s much as the elements of number theorycan have a complete an d consistent proof procedure. Th e com-pleteness noted in quantification theory, though attainable alsoin the elementary lgebra of real numbers (Alfred Tarski), ispossble neither in the elementary theory of whole numbers norin various other parts of mathematics; nor, afortiori, in the foun-

tainhead which is set theory.6. Further Aspec t s . The lack of a unique clear line in set theoryhs encouraged some (e.g., Hermann Weyl and at one point Rus-sell) to espouse constructionalism. The constructionalist in settheory stratifies the universe of classes into so-called orders, an dassumes '(3y)(x)(x t y .5= Fx)' only when ali quantified variablesother than 'y' ar e limited to orders lower than that assigned to 'y'.


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50 Se l ec t ed Logic Papers(Henr i Poincar: "predicative definition.") This course is safer ,more intuitive, and epistemologically more scrupulous than others,but it is inadequate to certain classical theorems in the theory offunctions of real numberstheorems which constructionalists areaccordingly prepared to abandon. An extreme variant of construc-tionalism is intuitionism (Liutzen E. J. Brouwer, Arend Heyting),which even revises elementary logic, abandoning the law 'p V p 'of excluded middle.

A di f er ent departure from truth-function logic is seen in moda llogic (Clarence L Lewis, Rudolf Carnap, Frederic B. Fitch), whichadmits, on a par with 'and', 'or', and 'not', the operator 'neces-sarily'. Against the need for this departure it hs been argued thatthe purposes thus served are better served by talking aboutformulas, attributing validity and tracing implications as hereto-fore in this article.

The standard elementary logic is also frequently departed f romin the purely exploratory spirit of abstract lgebranotably inmany-valued logic (Jan Lukasiewicz, J. Barkley Rosser, Atwell R.Turquette), whe r e the number of truth values is generalized f romtw o to n.

Apart f rom such substantive deviations, we can radically varythe mode of developmentof the standard sort of theory. For ex-ample, instead of taking membership and quantification as funda-mental, we may begin with identity, functional abstraction 'X,'(see above), and functional application. If by a harmless artficewe look upon sentences as names of their truth values, thenidentity and functional abstraction and application become asufficient basis fo r defining membership an d quantification (Frege,Church) and even the truth functions (Tarski). This alternativetrain of construction is less practical in some ways than that whichstarts with truth functions, quantification, and membership, butit brings added illumination. For functional abstraction can bes h o wn eliminable, in turn, in favor of a few specific functions,called combinators (Moses Schnfinkel, Haskell B. Curry). Wethus come out with a foundation fo r logic, including quantification,set theory, and their entire mathematical suite, which is devoid ofvariables.Proof theory is a domain drawn on but not described in the abovesurvey. It includes the proof s of the completeness and undecida-bility of quantification theory (Gdel, Church), the incompleta-bility of elementary number theory (Gdel), th e completeness of

L O G I C , B T M B O L I C 51elementary lgebra (Tarski); also the Lwenheim-Skolem theoremthat any consistent set of formulas of quantification theory is in-terpretable in the universo of whole numbers.

A power fu l device in proof theory is that of numbering ali themarks and strings of marks available in the notation of a theory,s as to obtain numerical relations parallel to the inferential re-lations between sentences of the theory concerned. It was by thusapplying number theory to the sentences of number theory, inparticular, that Gdel contrved to prove incompletability. Suchnumbering also underlay the discovery of the number-theoreticconcept of recursiveness, which makes precise sense of the idea ofmechanical computability (Herbrand, Gdel, Church, Alan M.Turing, Post, Stephen C. Kleene). The exact definition of "de-cision procedure" turns on recursiveness, as do the exact statementand proof of Church's theorem and Gdel's incompletabilitytheorem. Relevant also to the theory of machine computation,recursivenessis the worthy f ocus of a new branch ofnumber theory.

w.v. quine, encyclopedia american a, selected logic papers, symbolic logic, ii

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