[CS Dept., U Chicago]


Michael J. O'Donnell (Mike)

The Teacher

Lectures

Organization of Knowledge by Formal Systems

What do we mean by ``formal''?

The word ``formal'' is widely misunderstood, even by mathematicians who profess formalist foundations for their work. A good way to refine the understanding of formal is to consider its opposite. In common use, the opposite of formal might be ``casual,'' or ``relaxed'', or ``unrigorous.'' Mathematicians often call a derivation ``informal'' if it is incomplete or not described quite thoroughly. Mathematicians and other people also place formal in opposition to ``intuitive.'' None of these is a sensible opposite to formal for our purposes. Rather, the opposites we need are ``contentual'' and ``material.''

The word ``contentual'' is not normal English, but it is in the Oxford English Dictionary:

1. contentual
contentual , a. Philos. and Psychol. f. content sb.1 + -al on type of conceptual, eventual, etc., as tr. G. inhaltlich. Belonging to, or dealing with, content (opp. act or form).
  • 1909 W. M. Urban Valuation iv. 106 The decrease in these contentual aspects does not..mean loss in worth suggestion.
  • 1935 Brit. Jrnl. Psychol. XXVI. 208 As regards the `contentual' and the `actual' views, the former characterized the works of Sully and Volkmann.
  • 1962 W. G. Runciman Plato's Later Epistem. iv. 132 Distinguishing the formal from the contentual features of propositions.
I first saw ``contentual'' as a translation of the German ``inhaltlich'' in a paper by David Hilbert.

``Material'' is used quite commonly in English, so it gets a longer sequence of definitions. With some feeling of shame, I pruned them down to the ones that I need. You should look up the full list (give the command ox2 material on a campus Unix computer).

1. material
material , a. and sb. ad. late L. ma+teria+lis, f. ma+teria matter sb.1: see -al1.

NOTE: Cf. Fr. matriel adj. and sb., matriaux sb. pl., Sp., Pg. material, Ital. materiale; also the mod.Teut. forms (chiefly from Fr.), G. materiell adj., material sb., Dutch materieel adj., materiaal sb.

A. adj.

1. Scholastic Philosophy and Theol. (Opposed to formal.)

a. Pertaining to matter as opposed to form.


<lots of good stuff left out here>

2. Logic.

a. Concerned with the matter, as distinguished from the form, of reasoning. (Opposed to formal.)
<more wonderful stuff omitted, but I couldn't find ``material girl'' anywhere>

I intend to use ``material'' as ``pertaining to [the] matter'' of which something is composed. But the use of ``material'' in definition number 2 above is synonymous with ``contentual'': ``concerned with the matter'' that something denotes. Our language is mightily tangled.

In opposition to ``material'' and ``contentual,'' the appropriate sense of ``formal'' for our discussion of formal systems is

0. formal
referring only to the abstract forms of things, as opposed to their material composition or meaningful content.
By ``abstract forms'' I mean combinations of identities and observable correspondences between them. Identities are abstractions of things that we identify, ignoring their material composition and preserving only the distinction between references to the same identity vs. references to different identities. Observable correspondences are abstractions of relationships, ignoring the internal qualities of the relationships and the means by which they are observed, and preserving only the identities of things involved in the relationships. The preceding descriptions of identities and correspondences rule out material composition. In my definition of ``formal,'' I also explicitly rule out meaningful content denoted by the combinations of identities and correspondences. The explicit exclusion of meaningful content might not be logically essential to the definition. But the identitities and correspondences involved in interesting formal systems are usually understood as symbols and syntactic structures, so I emphasize that formal qualities have nothing to do with the meanings that those symbols and structures denote.

I gave my definition the number 0 because I had to make it up myself. Although the OED defines ``contentual'' as the appropriate opposite for ``formal,'' mentioning the opposition of content to form, and even quoting Runciman's use of the word ``formal'' in just the sense that we need, this comprehensive dictionary misses the appropriate definition of ``formal.'' But I quote the entire OED definition of ``formal below,'' to help clear up confusion between my use of formal and the others. Most of the time when people talk about ``formal mathematics,'' they appear to be using definition 6a, ceremonial. This tendency appears to be as strong in mathematicians as much as in nonmathematicians.

Definition 1, ``Of or pertaining to form, in various senses,'' is hard to argue with. The problem is, in what senses?

1. formal
formal , a. and sb.1 a. L. forma+lis, f. forma form sb.: see -al1. Cf. Fr. formel.

A. adj.

1. Of or pertaining to form, in various senses.

a. Metaph. Pertaining to the form or constitutive essence of a thing; essential. Opposed to material. So



formal cause
formal cause (see cause sb. 5).
  • C. 1386 Chaucer Melib. 433 The cause material been the fyve woundes of thy doghter. The cause formal is the manere of hir werkinge.
  • C. 1430 Art of Nombrynge (E.E.T.S.) 1 Sothely .2. manere of nombres ben notifiede; Materialle, as nombre is vnitees gadrede to gedres; Formalle, as nombre is a colleccione of vnitees.
  • 1447 Bokenham Seyntys Introd. (Roxb.) 1 The more clere That it may be the formal cause [He] Settyth in dew ordre clause be clause.
  • 1587 Golding De Mornay ii. (1617) 25 They be good, as in respect of their bare being: and euill, as in respect that they forgoe their formal being that is to say their goodnesse.
  • 1628 T. Spencer Logick 55 Man is..fit to loue the knowne good..this fitnes floweth from his reasonable soule or formall being.
  • 1669 Holder Elem. Speech 22 Of Letters the Material part is Breath and Voice; the Formal is constituted by the Motions and Figure of the Organs of Speech affecting Breath with a peculiar sound, by which each Letter is discriminated.
  • A. 1703 Burkitt On N.T. Luke xii. 32 The goodwill..of God is the original spring and formal cause, from whence all divine favours do proceed.
  • A. 1716 South Serm. (1744) X. 37 For deceit is the formal, constituent reason of hypocrisy.
  • 1814 Cary Dante, Par. ii. 71 Different virtues needs must be the fruits Of formal principles.
  • 1827 Whately Logic ii. v. Sect. 3 Whatever Term can be affirmed of several things, must express either their whole essence..or a part of their essence, (viz. either the material part, which is called the Genus, or the formal and distinguishing part, which is called Differentia).
b. Pertaining to the specific form of an animal or plant. Of a seed or germ: Endowed with a principle of form. Obs.
  • C. 1400 Lanfranc's Cirurg. 21 The foormal vertu which almyty god hath eue to the maris ordeyneth and diuidid euery partie of these spermes..til that the child be born.
  • 1605 Bacon Adv. Learn. ii. iv. Sect. 5. 19 Being as a plant that cometh of the lust of the earth without a formal seed.
  • 1677 Hale Prim. Orig. Man. iii. vi. 277 Although it be admitted that Insects and spontane orta do or may arise from a Semen or Principle that is not univocal or formal.
c. Pertaining to the outward form, shape, or appearance (of a material object); also, in immaterial sense, pertaining to the form, arrangement, external qualities (e.g. of a work of art, a composition, etc.). Also, occas. of knowledge: Theoretical as opposed to practical.
  • 1639 Evelyn Diary (1827) I. 15 Musick, in which I after--wards arriv'd to some formal knowledge, though to small perfection of hand.
  • 1655 G. S. in Hartlib Ref. Commw. Bees 27 Honey..out of which they [the Bees] doe separate a more fat substance, which they also transmute into Wax, with a formal transmutation.
  • 1837 Whewell Hist. Induct. Sc. (1857) I. 273 The distinction of formal and physical Astronomy.
  • A. 1853 Robertson Serm. Ser. iii. iii. (1872) 39 All living unity is spiritual, not formal.
  • 1860 Ruskin Mod. Paint. V. viii. i. 158 Invention Formal, otherwise and most commonly called technical composition.
d. Logic. Concerned with the form, as distinguished from the matter, of reasoning.


formal concept
formal concept tr. G. formaler begriff: a concept of logic, free from the descriptive content that would restrict it to any particular subject-matter (see quots.);



formal implication
formal implication (see quots.).
  • C. 1856 Sir W. Hamilton Lect. Logic xxvii. (1860) II. 64 The harmony of thought with the form of thought, is..Formal Truth.
  • C. 1856 Sir W. Hamilton Lect. Logic 231 App. i. xxvii. (1860) II. 64 The doctrine which expounds the laws by which our scientific procedure should be governed, in so far as these lie in the forms of thought, or in the conditions of the mind itself..may be called Formal, or Subjective, or Abstract, or Pure, Logic.
  • 1864 Bowen Logic ii. 42 All this, however, is but the elimination of Formal error.
  • 1870 Jevons Elem. Logic vii. 69 It is no part of formal Logic to teach us how to interpret the meanings of sentences.
  • 1903 B. Russell Princ. Math. Sect. 15 The relation of formal implication..holds between propositional functions when the one implies the other for all values of the variable.
  • 1922 tr. Wittgenstein's Tractatus 4. 126 In the sense in which we speak of formal properties we can now speak also of formal concepts. (I introduce this expression in order to make clear the confusion of formal concepts with proper concepts which runs through the whole of the old logic.)
  • 1932 Lewis Langford Symbolic Logic 101 The relation of formal implication is transitive, like material implication.
  • 1939 G. Ryle in Mind XLVIII. 149 To use the language of Kant and Wittgenstein, we were trying to treat formal concepts as if they were `proper' or material concepts.
  • 1939 G. Ryle in Mind 151 A formal concept is one which may have a place in a proposition about any subject-matter you please, and some..will be present in any proposition... `Not', `exists', `some', `other',..and many others are not peculiar to any special topics.
  • 1955 A. N. Prior Formal Logic 197 The Russellian `formal' implication is simply universal material implication.
  • 1965 E. J. Lemmon Beginning Logic iv. 154 The term `formal implication' was coined by Russell to describe the universal quantification over a material implication: i.e. a proposition of the form `(x) (FxGx)'.
e. Of or pertaining to customary form or conventionality.
  • 1712 Pope 1st. Ep. Miss Blount 42 Still in constraint your suff'ring Sex remains, Or bound in formal, or in real chains.
2. Characterized by, or regarded according to, its form; that is (so and so) in respect of form.

a. Theol.


formal sin
formal sin: one which is such in the full sense, as including not merely the outward act which is forbidden, but the circumstances which constitute it as sinful, e.g. evil intention. So


formal schism
formal schism,


schismatic
schismatic, etc. Opposed to material sin, etc.
  • 1641 J. Jackson True Evang. T. ii. 92 Therefore was there a positive Law..not to seeth the Kid in the mothers milke. Not that there was any direct, or formal sin, in that manner of Cookery.
  • ? 1656 Bramhall Replic. i. 66 Cannot God pardon formall, much more materiall Schism.
  • ? 1656 Bramhall Replic. ix. 341 They are not formall, but only materiall Schismaticks.
b. That is such in essence; strictly so called, essential. Obs.
  • 1691 Ray Creation ii. (1704) 289 The bottom of the Eye where the formal Organ of Vision is situate.
c. That is such merely in outward form or appearance. Obs.
  • 1581 Sidney Apol. Poetrie (Arb.) 35 Pretty Allegories, stealing vnder the formall tales of Beastes.
  • 1633 Earl Manch. Al Mondo (1636) 155 Formall penitents will easily part with so much of their sinne as may abate nothing of their profit.
  • 1634 Canne Necess. Separ. (1849) 231 The formal Protestants in England.
  • 1737 Bracken Farriery Impr. (1756) I. 286 It is a Kind of formal Leprosy which often begins in the Neck, Mane or Tail.
d. Of quoted statements: Exact with regard to form. Obs.
  • 1563 Foxe A. M. 708/1 What were the formall wordes, or at the least-wise in substance that I the sayde Bishop then vttered.
3. a. That is according to recognized forms, or to the rules of art or law. Formerly occas. const. to.
  • 1390 Gower Conf. III. 89 The wise man accompteth After the formal proprete Of algorismes a, be, ce.
  • C. 1425 Wyntoun Cron. ix. Prol. 56 Now Modyr of the Makare..To fair formale Fyne my labouris thow lede.
  • 1526 Pilgr. Perf. (W. de W. 1531) 168 The fourth condicyon of ye prayer of ye clene hert is, that it must be formall: that is to saye, it must be formed and ordred after the order of charite.
  • 1529 More Dial. Touchyng Pest. Sect Luther C ij b, A sylogysme resonynge, almoste as formall as is the argumente.
  • 1597 Morley Introd. Mus. 74 It followeth to speake of a formal closing without a dis-cord or Cadence.
  • 1602 Daniel Trag. Philotas iv. ii, And haue his Tryal formal to our Laws.
  • 1622 Malynes Anc. Law-Merch. 394 You may not say in the Bill, It may please you to pay..and most men will not vse the words (Make him good Payment) but the fewer words the more formall.
  • 1722 De Foe Col. Jack (1840) 235 No one place..could have held out a formal siege.
  • 1749 Fielding Tom Jones vi. vii. (heading) A picture of formal courtship in miniature, as it always ought to be drawn.
b. Made in proper form, regular, complete. Also in familiar use, `regular', unmistakable. Obs.
  • 1635 Earl Strafforde Lett. (1739) I. 410 An Indisposition which hath hindred me from writing..a formal Fit of the Gout.
  • 1673 Evelyn Mem. (1857) I. 89 We went..to see the formal and formidable camp on Blackheath, raised to invade Holland.
  • 1684 Lond. Gaz. No. 1953/1 Though the Lower Town has no other defence than a single Wall, yet his Highness found it convenient to make formal approaches to it.
  • 1719 De Foe Crusoe i. 247 As there was a Door or Entrance there into my Cave, I made a formal fram'd Door-case, and a Door to it of Boards.
c. Of a story, etc.: Elaborately constructed, circumstantial. Obs.
  • 1592 Warner Alb. Eng. vii. xxxv. (1612) 168 At full he could his lessons, and a formale lie would tell.
  • 1662 More Philos. Writ. Pref. Gen. (1712) 23 Such was that formal story of his casting incense on the Altar of an Idol.
  • 1663 Butler Hud. i. ii. 41 And never coyn a formal Lye on't, To make the Knight o'ercome the Giant.
  • 1708 Swift Sacr. Test Wks. 1755 II. i. 121 When the earl of Pembroke was in Ireland..a formal story was very gravely carried to his excellency.
4.

a. Regular, having a definite principle, methodical.

b. Of feature, stature, etc.: Regular, shapely.

c. Normal in intellect, sane. Obs.
a.
  • 1413 Pilgr. Sowle (Caxton) i. iii. (1859) 4 The ouer--most of the erthe was moost clere, and alwey the clerenesse amenussing dounward by verray formal processe.
  • 1701 Rowe Amb. Step-moth. v. ii. 2874 Formal Justice that severely strikes, And in an instant is serene and calm.
b.
  • 1525 Ld. Berners Froiss. II. clxiv. [clx.] 455 Therle of Foiz..was a goodly prince and of a formall stature.
  • 1548 Hall Chron., Edw. IV (an. 3) 194 b, She was a woman more of formal countenaunce, then of excellent beautie, but yet of such beautie and favor that [etc.].
  • 1576 Fleming Panopl. Epist. 377 Every joynte and limme..verie formall, and passing hansome.
c.
  • 1590 Shaks. Com. Err. v. i. 105 With wholsome sirrups, drugges, and holy prayers To make of him a formall man againe.
5. Done or made with the forms recognized as ensuring validity; explicit and definite, as opposed to what is matter of tacit understanding.
  • 1547 Boorde Brev. Health Preamble, Let him loke to it, and make a formal wyl or testament.
  • 1560-78 Bk. Discipl. Ch. Scot. (1621) 21 It hath power to excommunicate the obstinate, formall processe being led.
  • 1622 Bacon Holy War Misc. Wks. (1629) 127 As there are Formall and written Leagues, Respective to certaine Enemies; So is there a Naturall and Tacite Confederation, amongst all Men against the common Enemy of Humane Society.
  • 1626 Chas. I in Ellis Orig. Lett. Ser. i. III. 249 For Blennill he has yet but made his formale demands concerning the Ships.
  • 1771 Junius Lett. lxii. 321 Nor has there ever been a formal decision against them in any of the superior Courts.
  • 1838 Thirlwall Greece II. xiv. 228 Cleomenes, without waiting for a formal commission, immediately repaired to gina.
  • 1856 Froude Hist. Eng. (1858) II. vii. 129 Both the king and the archbishop had disobeyed a formal inhibition.
Predicatively of a law: Of unmistakable import, decisive. A Gallicism.
  • 1701 tr. Le Clerc's Prim. Fathers 260 He could not be transferred to Constantinople without breaking the Fifteenth Canon of the Council of Nice which is formal thereupon.
6.

a. Connected with or accompanied by form or ceremony; ceremonial, `state'.
  • 1602 Warner Alb. Eng. xi. lxvi. 283 Scarce Cleopatras Anthony was feasted with more cheere..than Jenkinson was heere: In formall Hawking, Hunting, Chace not them came Tristram neere.
  • 1841 Elphinstone Hist. Ind. I. xi. 355 The most general practice on formal occasions is [etc.].
  • 1875 W. S. Hayward Love agst. World 3 It will save the squire a formal call.
b. Of apparel: Ceremonial, proper to a dignity or office. Cf. formality 10. Obs.
  • 1593 Rites Mon. Ch. Durh. (Surtees) 43 [Pictures of bps. etc.] most largly and sumptuously sett fourth in there formall apparell.
  • 1656 Stanley Hist. Philos. iv. (1701) 136/1, I will not with a formal robe disgrace Myself.
7. That is merely matter of form:

a. Done or adopted for the sake of form or convention; perfunctory; having the form without the spirit or substance.

b. That is matter of routine only, not of substantial import.
a.
  • 1648 Milton Tenure Kings (1650) 45 A formal preachment huddl'd up at the odd hours of a lazy week.
  • 1676 Dryden Aurengz. ii. i, Of formal duty, make no more thy boast.
  • 1720 Watts Hymn, Come Holy Spirit, In vain we tune our formal songs, In vain we strive to rise.
b.
  • 1818 Cruise Digest (ed. 2) IV. 256 This doctrine..does not extend to mere formal acts.
8. Of persons, their manners and actions: Rigorously observant of forms; precise; prim in attire; ceremonious. Chiefly in reproachful use: Unduly precise or ceremonious, stiff.
  • 1514 Barclay Cyt. Uplondyshm. (Percy Soc.) 1 Amyntas was formalle propre in his gere.
  • 1594 Shaks. Rich. III, iii. i. 82 Thus like the formall Vice, Iniquitie, I morallize two meanings in one word.
  • 1596 Shaks. Tam. Shr. iii. i. 61 Are you so formall, sir?
  • 1607-12 Bacon Ess., Ceremonies (Arb.) 26 Especially they [Ceremonies] be not to be omitted to Straungers and formall Natures.
  • 1679 Penn Addr. Prot. i. vii. (1692) 27 He is reported Formal, that will not be Rude to Sacred Things.
  • C. 1689 Prior Ode, `While Blooming Youth' 25 Forc'd compliments and formal bows.
  • 1693 Hum. Conv. Town 125 The distant Justice of Peace, his formal Spouse, and Daughters.
  • 1711 Addison Spect. No. 119 5 To make Conversation too stiff, formal and precise.
  • 1749 Chesterf. Lett. (1792) II. cxciii. 220 All the evening in formal fine company.
  • 1853 Lytton My Novel ii. vii, More familiar with his master than we formal English permit our domestics to be.
9.

a. Marked by extreme or excessive regularity or symmetry; stiff or rigid in design; wanting in ease or freedom of outline or arrangement.
  • 1597 Shaks. Lover's Compl. 29 Her hair, nor loose, nor tied in formal plat.
  • 1753 Hogarth Anal. Beauty vi. 34 When any part of dress has not the excuse of fitness or propriety for its uniformity of parts, the ladies always call it formal.
  • 1807-8 W. Irving Salmag. (1824) 116 Your plaited shirts, Your formal bag-wigs.
  • 1873 Black Pr. Thule xxi. 353 Small windows with formal red curtains.
  • 1874 L. Stephen Hours in Library (1892) I. iii. 120 Pope..was one of the first..to break through the old formal school of gardening.
b. In immaterial sense: Having a `set' or rigorously methodical aspect or character.
  • 1726 Shelvocke Voy. round World (1757) 423 It would perhaps be too formal to enter upon a discourse concerning their government.
  • 1846 McCulloch Acc. Brit. Empire (1854) II. 367 Formal harangues of this sort are about the least efficient of all the modes in which information can be conveyed to the student.
  • 1865 Grote Plato (1875) Pref. 5 The dramatic-as contrasted with the formal and systematising.
B. sb. A thing that is formal; esp. (orig. and chiefly U.S.) (an) evening dress; an engagement at which such dress is worn.
  • 1605 Timme Quersit. ii. iv. 14 Simples may be distinguished into those things which are simple formes, and into those which are simple matters; or into those things which are simply formals, and into those which are simply materials. Those things which are simply formall are astrall and spirituall.
  • 1875 Whitney Life Lang. v. 95 They are etherealized formals.
  • 1941 Amer. Speech XVI. 96 They are all right for casuals or spectator wear, but I'd get that formal in either Waltz Blue or Coronado Coral.
  • 1946 New Yorker 28 Sept. 27/2 The neighboring children were strange, with..their queer talk of dates, and formals, and going steady.
  • 1948 New Yorker 27 Mar. 29/1 The initiates wore evening dresses (formals, formals, Emily reminded herself to say).
  • 1961 M. Beadle These Ruins are Inhabited (1963) v. 71 If a dinner invitation does not specify `informal', guests assume that black ties will be worn. Young ladies then wear short formals, and old ladies wear floor-length evening dresses.
  • 1968 A. York Predator vi. 86 She looked sufficiently virginal to be attending her first formal.


2. formal
formal, var. form of formel.


3. formel
'formel, formal formal, sb. Obs. Also 4-5 formaylle, 7 formale, fore-male. a. Fr. formel adj. (see formal), which occurs in faucon formel, and latinized as formelus in a letter of Magnus of Norway to Edw. I, as an epithet applied to hawks. As the female hawk was greatly superior for purposes of sport, the sense of formel in this application may be `regular', `proper' (see formal a.). NOTE: Cf. Fr. forme, `a term of Hawking, the female of a bird of prey that gives its name to the species' (Littr). The female of the eagle or the hawk. Also attrib.
  • C. 1381 Chaucer Parl. Foules 371 To chese or for to take, By hir acord, his formel or his make.
  • C. 1381 Chaucer Parl. Foules 373 Nature held on hir honde A formel egle.
  • ? A. 1400 Morte Arth. 4003, I salle neuer..ffawkone ne formaylle appone fiste handille.
  • A. 1605 Montgomerie Misc. Poems xviii. 38 Quhilks vhen they sau they wroght in vane, The formels fair auay they fure.
  • 1616 Surfl. Markh. Country Farme 712 Of Merlins there are both males and females..the female is called the formale.
  • 1674 N. Cox Gentl. Recreat. ii. (1677) 215 If you will fly with a Merlin at a Partridge, chuse the Formal, which is the Female. The Jack is not worth the training.
  • 1688 R. Holme Armoury ii. 239/1 Fore-Male, the term given to all Females of Hawks.

    • Put another way, a formal system is a system whose definition is entirely about form, rather than one that is expressed in the right form or following a particular form. Furthermore, the sort of form involved in the definition of a formal system is abstract relational form, having to do with postulated relations between identities, rather than geometric form, or some other concrete rendition of form.

    Copyright Michael J. O'Donnell <michael_odonnell@acm.org>.
    Licensed for free use.

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