A C T A U N I V E R S I T A T I S L O D Z I E N S I S
FOLIA PH I L.OSOPH 1C A 9. 1993
Jan W oleński
ANALYTICITY AN13 M ETAM A TH EM ATICS
The distinction o f analytic/synthetic was explicitly stated for the first time by K a n t w ho referred it to judgem ents. O th er a u th o rs applied the distinction also to sentences propo sitio n s and statem ents; in w hat follows. I shall use the form "sentence' even if reviewed a u th o rs em ployed a n o th e r nam es. Before K an t, related ideas concerned with the distinctions a prio ri/a posteriori and necessary/contingent had been developed m ainly by H um e and Leibniz. A lthough p rc-K an tian s did not used the term s 'a n aly tic ' and "synthetic’, it is com m on to regard L eibnitian definition o f necessary tru th (as a sentence true in all possible w orlds) o r H um ean treatm ent o f relations between ideas (as recorded by tautologies) as im p o rtan t p roposals conccrning the concept o f analyticity.
F or K ant, the linguistic stru ctu re ,,A is B" is the general form o f sentence. N ow a sentence S is an analytic if an d only if its predicate A is ‘co n ta in e d ’ in its subject B; otherw ise S is a synthetic sentence. It follows from K a n t’s definition th a t negations o f analytic sentences are self-contradictory. M oreover, analytic tru th s arc uninform ative (tauto lo g o u s) becausc they merely analyse the relevant subject concept. F orm al logic for K an t consists o f analytic sentences. On the o th er hand, synthetic sentences consist in a synthesis o f concepts and provide an inform ation. All analytic sentences are for K an t a priori by definition but synthetic ones can be either a priori o r a posteriori. The celebrated problem o f K a n t's philosophy concerned the possibility o f sentences which w ould be are b o th synthetic and a priori. K a n t him self was entirely convinced th a t such sentences exist.
The p o st-K an tian philosophers proposed m any definitions o f analyticity. Several o f them are included in the following list1 (analytic = analytically true):
( l a) S is analytic iff S is true in all possible worlds; ( l b) S is analytic iff S could not be false;
(l c) S is analytic iff not-S is self-contradictory;
( I d) S is analytic iff S is tru e by virtue o f m eanings and independently of facts;
(l c) S is analytic iff cither S is logically true o r S can be turned into a logical tru th by puttin g synonym s for synonym s;
(If) S is analytic iff S com es ou t true under every state-description; (l g) S is analytic iff S can be reduced to logical tru th by definition; ( l h) S is analytic in a language L iff S is true according to the sem antical rules o f L.
T he definition ( l a) goes back to Leibniz, ( l b) a nd (l e) are m entioned as possible explications by Q uine in his very fam ous criticism o f analyticity2 (l c) is proposed by S traw so n 1, ( I d) records a typical positivistic treatm ent o f analyticity4, (If) a nd ( l h) are tak en from C a rn a p 5, and (l g) expresses Frege’s definition o f analyticity.
V arious general logical term s o ccur in definitions ( l a) (l h). I ru th , logical tru th , definition an d co n tra d ictio n a p p e a r explicitly in them b u t o th er, lor instance m odel, provability o r consistency implicitly via possible w orlds (state descriptions), logical tru th and co n trad ictio n respectively. We can rewrite for instance ( l a) and (If) as
(2) S is analytic iff S is true in all m odels
and ( l g) as (note th at (3) is closer to Frege’s original form ulation than (lg )) (3) S analytic iff S is provable exclusively by logic and definitions. Im p o rtan t aspccts o f m etalogical conccpts like tru th , consistency or provability are form ally regulated by m etam athcm atical theorem s; for sim p-licity, 1 assum e th at m etam athem atics com prises m etalogic and form al sem antics. So we can ask w hat follows from m etam athem atics for the ..philosophy o f analyticity” . M y aim in this p aper is to put together (with some com m ents) various observations on analyticity which have been m ade by several con tem p o rary logicians from the m etam athcm atical point o f view.
I shall center on so called lim itative theorem s, in p artic u la r
(4) if X co n tain s form alized P eano arithm etic, then X is incom plete if consistent (the first G ödel incom pleteness theorem );
2 See: V. v a n Q u i n e , Two dogmas o f empiricism, ..Philosophical Review" 1951, N o. 60, p. 20-43.
3 See: P. S t r a w s o n , Introduction to Logical Theory. M ethuen, London 1952, p. 21. 4 See: A. A y e r , Language, Truth and Logic, Penguin Books, Harmodwordth 1971. p. 104-106.
s See: R. C a r n a p , Logical Foundations o f P robability, Routledge and Kegan Paul, London 1962, p. 83 and R. C a r n a p , Meaning and Necessity. The University o f Chicago Press. Chicago 1956, p. 8, 10.
(5) if S contains form alized P cano arithm etic, then consistency o f S is im provable in S (the second G ödel incom pleteness theorem );
(6) P eano arithm etic and first o rd e r logic arc not decidable (the C hurch undecidability theorem ).
The first analysis o f analyticity with the help o f m etam ath em atics was given by C a rn a p ft. He distuinguished L anguage 1 consisting o f elem entary logic together with the p o rtio n o f arith m etic sufficient for arith m ctizatio n (in the sens o f G ödel) and L anguage II which co n tain s all m eans which are needed for expressing classical m athem atics in it. N ow analyticity in Language 1 is defined by
(7) S is analytic in LI iff S is a consequence o f the null class o f sentences (o r every sentence);
H ow ever, (7) is to o n arrow for LII because arithm etic is incom plete w hat causes that ,,in every sufficiently rich system for which the m ethod o f d erivation is prescribed, sentences can be co n stru cted which, th o u g h they consist o f sym bols o f the system , are yet not resoluble in accordance with the m ethod o f the system th a t is to say, are neither dem o n strab le n o r refutable in it. A nd. in particu lar, for every system in which m athem atics can be form ulated, sentences can be constructed which are valid in the sense o f classical m athem atics but not d em onstrable w ithin the sy stem ''7. So we have sentences which are no t consequences o f every sentence. T o solve this difficulty, C a rn a p (he w anted to have all m athem atical tru th s am o n t analytic sentences) proposes to adm it infinite sets o f prem ises and supplem ent rules o f p ro o f by non-effective ones, for instance oj-rule. C a rn a p ’s definition o f analyticity for Language II is to o com plicated in o rd e r to present it here in a detailed way but the general idea is cap tu red by
(8) S is analytic in LII iff S is derivable from analytic sentences by rules o f p ro o f which arc adm issible in LII.
As far as I know, G ödel adressed to the problem o f analyticity only once in his published w orks, nam ely in his paper on R ussell's m athem atical logic8. A ccording to G ödel
(9) S is analytic iff A is a special case o f the law o f identity in virtue o f explicite definitions o f term s o r rules o f their elim ination.
However. G ödel observes that (6) implies non-analyticity o f arithm etic. A dm itting sentences o f infinite length does not save the situation because to prove th at some im p o rta n t m athem atical theorem s (for instance, the axiom o f choicc) arc analytic, one w ould have to assum e analyticity o f the whole
0 See: R. C a r n a p . Logical S vn lax of Language, Routledge and Kegan Paul. London. 7 Ibid., p. 100.
8 See: К G ö d e l . R ussell’s M athem atical Logic, (in:) The Philosophy o f Bertrand Russelll, ed. P. Schilpp. Open Court. La Salle 1944, p. 123 153.
m athem atics in advance. G ödel also considers a n o th e r definition o f analyticity, nam ely a version o f ( I d) b u t he does no t link it with an y m etam athcm atical fact.
The next step in the history in q uestion was m ade by C opi an d T u rq u e tte 9. Copi exam ines the follow ing definition o f analyticity
(10) S is analytic iff its tru th o r validity follows from the syntactical or gram m atical rules governing a language rules in which it is expresed. from the p oint o f view (4) form ulated by him as
(11) given any reasonably rich language, there is lion-em pirical, non-inductive prop o sitio n expressible w ithin it which is not decidable on the basis o f the syntactical rules o f that langauge.
T hen C opi says th at ( I I ) leads to
(12) there arc a priori non-analytic tru th s,
which destroys the analytic theory o f a priori (all a priori sentences are analytic).
T u rq u e tte m akes several objection against C opi. Let me m ention two. The first is general: „In fact, the claim that there are G ödel synthetic a priori tru th s then am o u n ts to nothing m ore th an a restatem ent in m isleading philosophical language o f som e w ell-established logical results, n otably o f what is usually called G ö d cl's second incom pleteness th e o re m " 10. Secondly. T u rq u e tte o bser-ves th a t undecidablc statem ents could be interpreted as em pirical or w ell-form ed but devoid o f m eaning.
Copi in his a n sw e r" says th a t his theses are not derived from undecidablc sentences but from (he fact th at „th ere are such statem ents as G ó d e ľs which arc a priori tru e but no t an aly tic” 12. M oreover, he rejects the em pirical theory o f m athem atics and observes th at regarding undecidable sentences as devoid o f m eaning is untenable because wc understand them .
T u rq u e tte positive solutions require either accepting th at m athem atics is em pirical or a revision o f logic: both proposal m ust meet several well-know n objections. T u rq u e tte 's general objection against C opi raises a serious m et-hodological problem . G ö d e l’s theorem s (like o th e r lim itative results) says nothing on analyticity or apriority. So C o p i’s form ulation o f (4) is in fact its certain philosophical in terp re tatio n which should be separatately justified. M oreover, (12) is derived by C opi from (10) but it m ay not hold u n d er o ther definitions o f analyticity.
“ See: !. C o p i. M odem Logie and the Synthetic a priori, „Journal o f Philosophy" 1949, N o. 46. p. 243-245; l. C o p i, G ödel and the Synthetic a priori: a Rejoinder, „Journal o f Philosophy" 1950, N o. 47, p. 633-636; A. T u r q u e t t e , G ödel and the Synthetic a priori, „Journal o f Philosophy" 1950. N o. 47. p. 125-128.
10 T u r q u e t t e . Gödel.... p. 126. 11 See: C o p i . Gödel...
K cm eny’3 argues th at the concept o f intended m odel (in terp retatio n ) form s an ad e q u ate conceptual base for form al sem antics. A ssum e th at we define analytic propositions as those which are universally valid, i.e. hold in all models. This definition is too narro w in virtue o f incom pleteness o f arithm etic. Kcmeny argues th at a m ore satisfactory account o f analyticity is to be obtained with the help o f the concept o f intended m odel.
Let T be a theory, i.e. a set o f sentences closed u n d er the consequence o peration. N ow all intended m odels o f T have exactly the sam e universes. M oreover, if M and M ' are intended m odels, then both can m utually differ only with respect to valuation o f extralogical constants; K cm eny considers arithm etical co n stan ts as logical. Then
(13) S is analytic in L iff A is tru e in all L -interpretations, i.e. L intended m odels o f L.
Assum e that S is analytic in L. A theory T is com plete (K cm eny says th at it is the m ost n atu ral concept o f com pleteness) if and only if
(14) S belongs to C n(T ) if A is analytic in T.
If T is com plete, then its analytic tru th s can be defined as valid in all models. But if T is incom plete, this definition m ust be replaced by (13) because for instance we have arithm etical tru th s which arc not valid in all m odels o f arithm etic.
K em eny's ap p ro ach raises som e doubts. Let S be an undecidable form ula in its intended m eaning. C o n sider its negation not-S. We can easy define a set o f m odels in which S holds. O ne can even claim th at m odels o f not-S (not those o f S) arc intended. T his m eans th a t not-S is analytic on this claim. So we o btain th a t two m utually co n tiad icto ry sentences are analytic. This reasoning shows that the concept o f analyticity via L -interpetations is ra th e r pragm atic and relativised than sem antic an d absolute.
B orkow ski14 considers two definitions o f analyticity, nam ely
(15) S is analytic in the syntactic sense iff S is provable exclusively by logic; (16) S is analytic in the sem antic sense iff S is true in all models. A ccording to B orkow ski, the first G ödel theorem implies th a t not every sentence sem antically analytic is also syntactically analytic. H ow ever, this thesis is dubious. If sentence S is true in all m odels, it is (by com pleteness theorem ) provable exclusively by logic. This m eans th a t both classes o f analytic sentences m utually coincide.
See: J. K e m e n y . A New Approach ю Semantics, ..Journal o f Sym bolic Logic" 1956. No. 21. Part 1, p. I 27; Part 2, p. 149 161.
14 L. B o r к о w s к i. Deductive Foundation and Analytic Propositions. „Studia Logica" 1966. N o. 19, p. 59 72; L. B o r k o w s k i , Logika form alna, PW N, Warszawa 1970; L. B o r k o w s k i .
Wprowadzenie do logiki i teorii mnogości, Towarzystwo N aukow e K atolickiego Uniwersytetu
D c L o n g 15 argues th at (the form ula C on(A r) m eans „arith m etic is consis-te n t” ).
(16) T he sentence C on(A r) under its intended in terp re tatio n is synthetic
a priori.
T he form ula expressing consistency o f arithem ic is synthetic because it is not provable exclusively by general logic an d definitions an d a priori because il arithm etic is consistent, it is necessarily so.
N ow assum e th a t C on(A r) is necessary true. Let M is the stan d ard m odel o f Ar. So N ec(C on(A r)) is true iff and only if C on(A r) is true in all m odels accessible from M. However, these m odels are not determ ined a priori bu t with respect to pragm atic criteria o f standardcness. T o o btain (16) one has to show th at C on(A r) holds in all m odels in which P eano axiom s hold but it would be inconsistent with undecidability o f C on(A r).
C asto n g u a y 16 claim s th a t C h u rc h ’s theorem (together with C h u rch 's thesis) implies th a t m athem atical know ledge is synthetic a priori. H ow ever, this is too strong claim because (6) implies only th a t m athem atical know ledge is nor reducible to purely algorithm ic procedures. C astonguay seems to assum e
(17) if X is a set o f analytic sentences, th an X is decidable. But this supposition is by no m eans obvious.
T here is not system atic treatm ent o f analyticity from the point o f view m etam athem atics. On the o th er hand, m etam athem atical seem to be o f a fundam ental im portance for any analysis o f analyticity. Let me finish this survery with som e very general o b serv atio n s17. M etam athem atics suggests two divisions o f analytic sentences: (I) into syntactic, sem antic and pragm atic (note how ever th a t my proposals in this respect considerably differ from those o f B orkow ski18), an d (II) into absolute and relative. T he proposed definitions are as follows:
(17) S is an absolute sem antic analytic sentence iff S is universally valid; (18) S is an absolute syntactic analytic sentences iff S is an ab so -lute sem antic analytic sentence and S belongs to a decidable set ol logical truths;
(19) S is a relative sem antic analytic sentence in a theory T iff S is tru e in all m odels o f T;
(20) S is a relative syntactic analytic sentence in a theory T iff S is a relative syntactic analytic sentence in a theory T and S belongs to a decidable set o f tru th s o f T;
15 See: H. D e L o n g , A Profile o f M athem atical Logie, Addison-W esley, Reading, Mass. 1970.
16 See: Ch. C a s t o n g u a y , Church’s Theorem and the Analytic-synthetic Distinction in
M athem atics, „Philosophica" 1976. N o. 18. p. 77 89.
11 See: J. W o l e ń s k i , M etam atem alyka i epistemologia, PW N. Warszawa (forthcoming). 18 See: B o r k o w s k i , Deductive Foundation...
(21) S is a p ragm atic analytic sentence in a theory T iff S is tru e in all stan d ard m odels o f T.
Obviously we have,
(22) absolute syntactic analytic sentences £ absolute sem antic analytic sentences <= pragm atic analytic sentences (the sam e holds if'a b s o lu te ' will be replaced by ‘relative’).
So syntactic analytic sentences are those which can be resolved by algorithm ic m ethods. M oreover only logic consists o f absolute analytic sentences. These consequences are consistent with m any trad itio n al accounts concerning analytic sentences.
Institute o f Philosophy Jagiellonian University Poland
Jan W oleński
A N A L IT Y C Z N O Ś Ć I M E T A M A T E M A T Y K A
Chociaż rozróżnienie sądów analitycznych i syntetycznych pojaw iło się po raz pierwszy u Kanta, to pokrewne pojęcia można odnaleźć już u Hum e’a i Leibniza. A utor zestawia i analizuje różne definicje i charakterystyki pojęcia analityczności, jakie proponowali m. in.: Kant. pozytywiści. Frege, Carnap. Strawson i Quine. Wskazuje się, że w badaniach nad zagadnieniem analityczności często odw oływ ano się d o takich pojęć m etalogicznych, jak. prawdziwość, mesprzeezność. czy dow iedlność, a te z kolei zostały scharakteryzowane na gruncie metamatema- tyki przez tzw. twierdzenia limitacyjne, w szczególności przez twierdzenia Godła o niezupetności i twierdzenie Churcha o nierozstrzygalności. W związku z tym referowano dyskusję nad związkiem ww. twierdzeń z zagadnieniem rozstrzygalności prowadzoną przez sam ego G odła, a także przez Turquette'a, C opiego. Kem eny’ego. Borkowskiego i in.
