Abstract
The theory of the “Quantification of the Predicate” attempts to transform the traditional logic of the four categorical forms (Every S is P; No S is P; Some S is P; Some S isn’t P) into a system of eight or even twelve propositions in which the simple predicate P is replaced by a quantified predicate like ‘some P’, ‘every P’ and perhaps even ‘no P’. According to the standard historiography of logic, such a theory was invented in the 19th century by W. Hamilton and Augustus De Morgan. However, already in the 17th century, the Spanish logician Juan Caramuel y Lobkowitz published a book “Theologia rationalis” in which propositions with quantified predicates are systematically investigated. By way of a remarkable extension of the traditional theory of conversion, Caramuel arrives at a system of logical inferences which might be considered as a forerunner of Hamilton’s theory. However, Caramuel’s “method” basically consists only in listing various examples of true and false propositions. Therefore, his theory fails to provide a general semantics for propositions with a quantified predicate. One variant of such a semantics was developed in the 18th century by Gottfried Ploucquet. Another completely different one had been sketched already in the 17th century by Gottfried Wilhelm Leibniz.
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- 1.
Cf. [11, p. 279]: “All A is all B, Any A is not any B, All A is some B, Any A is not some B, Some A is all B, Some A is not any B, Some A is some B, Some A is not some B”.
- 2.
Cf. [7, pp. 297–319].
- 3.
Cf. [26, p. 9], where Venn constrains that “neither Mr. Bentham nor Mr. Solly seem to me to have understood exactly the sense in which their scheme was to be interpreted”.
- 4.
Cf. my contribution to the Montreux conference on the Logical Square [14].
- 5.
Quoted according to the German Wikipedia article “Juan Caramuel y Lobkowitz”, online access on January 17, 2014: “Allerdings behielt kaum etwas davon dauerhafte Bedeutung.”
- 6.
Cf. the Online-bibliography organized by Jacob Schmutz http://web.archive.org/web/20070703162930/http://www.ulb.ac.be/philo/scholasticon/bibcaramuel.html.
- 7.
Cf. [25, p. 274]: “La teoriá de Caramuel sobre la cuantificación del predicado no sólo es anterior a la de Hamilton sino que es más completa y sistemática.”
- 8.
Cf. Theol, p. 218. In addition to this “subalternation of the subject”, Caramuel also admits of a “subalternation of the predicate”: “Aliquod animal est omnis homo; Ergo aliquod animal est aliquis homo […]” (Theol, p. 73).
- 9.
Cf. Theol, p. 69: “Duae contrariae possunt et solent esse simul falsae: non simul verae. Duae subcontrariae solent esse simul verae, non simul falsae. Duae Contradictoriae, nec simul falsae, nec simul verae.”
- 10.
At the beginning of § DCLXX, Caramuel defines conversion as a truth-conserving commutation of subject into predicate and predicate into subject, of which there exist three types: “Primò simpliciter, Secundò per accidens, Tertiò per contrapositionem”.
- 11.
Again, Caramuel doesn’t state these laws in an abstract form but only by way of examples. “Universalis negativa converti potest simpliciter, ut si diceres Nullus homo est argenteus: Nullum argenteum est homo […] Particularis affirmativa in omni materiâ convertitur simpliciter; ut Aliquis homo est album; Aliquod album est homo” (Theol, p. 219).
- 12.
Cf. § DCLXXI: “PRIMO igitur Universalis affirmativa […] in omni autem materiâ potest converti per accidens. Ut Omnis homo est animal: Aliquod animal est homo. Omnis homo est albus; Aliquod album est homo.”
- 13.
E.g. “SECUNDO, Universalis negativa […] poterit etiam converti per accidens: ut Nullus homo est lapis: Aliquis lapis non est homo”.
- 14.
For SeP → PeS by Conv 1 and PeS → PoS by Sub 1, hence Conv 4. Similarly, SaP → SiP by Sub 2 and SiP → PiS by Conv 2, hence Conv 3.
- 15.
A more powerful proof of the invalidity of Conv 6 consists in the observation that, if Conv 6 were valid, then Conv 4 would become provable. For assume that SaP, then, because of Opp 1, ¬SoP. Now if Conv 6 were valid, then PoS would entail SoP; hence, by logical contraposition, ¬SoP would entail ¬PoS, so that (again by Opp 1) we would obtain PaS.
- 16.
“CONTRAPOSITIO afficit omnes propositiones & est quaedam Conversio, quae servat propositionis quantitatem, convertit extrema, & utrumque infinitat” (§ CCXLV, p. 73).
- 17.
“Sed hae barbarae locutiones sunt, à quibus debemus abstinere; quia Dialecticus debet saltem ita loqui, ut ab audientibus intelligatur.” More appropriate examples for Contra 4 may, however, be found in other places of Theol, e.g. p. 73, § CCXLV: “Aliquis homo non est leo: Ergo aliquod non leo non est non homo”.
- 18.
“Et tandem, […] universalem affirmativam, & particularem tam affirmativam quam negativam [!] converti per contrapositionem.”
- 19.
Cf. p. 69: “AEQUIPOLLENTIA est Enunciationum aequivalentia: haec enim videntur idem significare […] Nullus homo est lapis. Omnis homo non est lapis. Omnis homo est non-lapis.”
- 20.
A general proof of Obv 2 and Obv 3 is easily obtained from Obv 1, Obv 4 by means of the laws of opposition, e.g. \(\mbox{SoP} \rightarrow \neg\mbox{(SaP)}\ (\mbox{by }{\rm O{\scriptstyle PP}\ 1}) \rightarrow \neg(\mbox{Se}\sim\mbox{P})\ (\mbox{by } {\rm O{\scriptstyle BV}\ 4}) \rightarrow \mbox{Si} \sim\mbox{P}\ (\mbox{by } {\rm O{\scriptstyle PP}\ 2})\).
- 21.
Obv 4 easily follows from Obv 1 by substituting ∼P for P.
- 22.
Cf. Theol, p. 221, where Caramuel first explains two laws of “Logicae Metamorphoseos”: “Prima, Propositio asserens, cuius praedicatum est infinitum, converti in negativam potest […] verbi gratia [ex] hâc Homo est non equus […] erit Homo non est equus […] Secunda, Cuiuscumque propositionis negativae negatio transeat ad praedicatum […] ut patet in hâc Petrus non est canis, quae transformatur in hanc, Petrus est non-canis.” With the help of these laws he is then able to transform ‘Petrus non est non homo’ as follows: “Respondeo debere juxta secundam regulam ac proptereà in hanc, Petrus est non non-homo, sive quod idem est, Petrus est homo;”
- 23.
Cf. Theol, p. 219: “Hae dicta sunt ad mentem Veterum sed & placet adhuc penitius hanc difficultatem contemplari”.
- 24.
Cf. [1, p. 183]: “L’attribut d’une proposition affirmative […] est toûjours consideré comme pris particulierement […] L’attribut d’une proposition negative est toûjours pris generalement”.
- 25.
In English, the UN might be paraphrased as ‘No S is any P’ or also as ‘Every S isn’t any P; and the PN is quite naturally formulated as ‘Some S isn’t any P’. Here then the predicate may be considered as taken “universally”. The Latin counterpart of ‘any’ would be ‘ullus’, but Caramuel nowhere considered this quantifier.
- 26.
The paragraph starting at the bottom of the right column of p. 219 bears the headline “De Transsubstantiatione materiali”. While Caramuel normally writes the number of a § on the margin, # DCLXXIII is missing here!
- 27.
In § CCCCLXVIII, Caramuel defines an indefinite proposition as “one which has an undetermined quality, i.e. which has no quantifier expression”. And he goes on to explain that, e.g. ‘Homo est animal’ is sometimes equivalent to a universal and sometimes equivalent to a particular proposition.
- 28.
“Maioris claritatis gratiâ propositiones veras à falsis distinxi charactere: illas enim scripsi cursivo, has romano.” Pastore [18] appears to have overlooked this important point even though Caramuel had written on the margin: “Nota id bene.” Otherwise Pastore’s reconstruction of the theory of transubstantiation consists only in listing the 16 possible combinations of quantities without reproducing also the corresponding propositions. Furthermore, he entirely dismisses § DCLXXIV which, allegedly, “only repeats the list of the 16 quoted cases”. Pastore overlooks that Caramuel here deals with the conversion of negative propositions!
- 29.
“Ex dictis constat ubi & quando possint propositiones transsubstantiari servata veritate; ac ideo non est opus speciales regulas multiplicare.”
- 30.
Cf. the discussion of principle TransSub 3 in the previous section.
- 31.
Cf. the passage from Theol p. 69 (already quoted in footnote 22) where Caramuel transforms ‘Nullus homo est lapis’ into ‘Omnis homo non est lapis’.
- 32.
Cf. Theol, p. 221: “Haec est Conversio totalis affirmativarum [pro]positionum; notavi bonas & malas illationes, ut possis uti discretione, si velis.”
- 33.
This assignment of truth-values nevertheless leaves open the possibility that the general inference scheme is valid.
- 34.
This is also the sense in which Hamilton [11] interprets his formula ‘All S is all P’.
- 35.
Cf. Theol p. 221: “Videamus igitur, quo [modo negativae] convertantur”.
- 36.
All these problems have been overlooked by Pastore who thinks that Caramuel here only “repeats the first twelve schematic letters, together with corresponding examples, of the preceding §.” Cf. [18, p. 131].
- 37.
As usual, the other inferences involving quality I or S are left out of consideration here.
- 38.
This counterexample escaped the attention of Caramuel because he forgot the negation ‘non’ and thus considered instead the conversion of the (true) PA ‘Aliquod animal est aliquis homo’ into the (equally true) PA ‘Aliquis homo est aliquis animal’.
- 39.
Cf. § DCLXXXII: “Quodsi pronomen hoc sit individuale et singulare […] utraque propositio erit falsa: non enim reperitur aliquod animal singulare, cui idemtificetur omnis homo”.
In the previous §, Caramuel had already pointed out that under the individualistic interpretation even proposition ‘Homo est omnis homo’ is false: “[…] quia pro nullo suppositorum verificari potest. Neque enim Sortes est omnis homo, neque Plato, neque aliquis alius.”
- 40.
Cf. Theol, p. 222: “[…] si pronomen illud sumitur pro ecceitate, aut potius taleitate specificâ […] hoc animal, signato homine ut sic, erit idem ac animal rationale: & hoc sensu utraque propositio erit vera.”
- 41.
Cf. § DCLXXXI: “Hanc, Omnis homo est aliquod animal aio transire posse in hanc, Aliquod animal (nimirum rationale, sumendo illud Aliquod pro specie vaga) est omnis homo”.
- 42.
Cf. Theol, p. 222: “At Ego dico in hâc propositione, Aliquod animal est omnis homo illud Aliquod vel sumi pro specie vagâ, vel pro individuo vago: & addo eandem in primâ acceptione esse veram, & in secundâ falsam”.
- 43.
As the reader may verify, also TotalConv 2–4 and TransSub 1, 4 become valid under the present set-theoretical interpretation, while TransFig 1, 2 and TransSub 2, 3 are invalid.
- 44.
Caramuel would even consider the two universal propositions ‘Omnis lapis est nullus homo’ and ‘Omnis leo est nullus homo’ as true. Hence, given the above truth-conditions, it would follow that the set of all lions is identical with the set of all stones!
- 45.
Somewhat surprisingly, this formula is even compatible with the stronger proposition ‘Omnis S est omnis P’. If S=P is any set with at least two individuals {x,y}, then choosing Z={x} and Y={y} shows that ∃Y∃Z(Z⊆P∧Y⊆S∧Y≠Z).
- 46.
Cf. [18, p. 133]: “La grande preoccupazione del Caramuel è appunto il calculo e per esso l’esaustione di tutti i casi possibili”.
- 47.
Cf. [19, p. 3]: “O. praefixum denotat omnitudinem positive sumtam […] Q. vel q. praefixa denotant particularitatem […] S−P denotat: S est P. S>P denotat: S non est P”. In some places he also uses ‘N.P’ for the negative quantifier ‘Nullus P’.
- 48.
Note also that Ploucquet’s system fully accords with the traditional doctrine that the predicate of an affirmative proposition is always particular while the predicate of a negative proposition is always universal.
- 49.
- 50.
Note, incidentally, that Leibniz commits a fallacy when he maintains that PAQP might be expressed by “Omne P esse quoddam S”. The latter proposition is a UA and thus it has to be formalized as ∀x(Px→∃y(Sy∧y=x)). As modern logic shows, however, one may not simply interchange the two quantifiers within such a formula!
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Lenzen, W. (2015). Caramuel and the “Quantification of the Predicate”. In: Koslow, A., Buchsbaum, A. (eds) The Road to Universal Logic. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-10193-4_17
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