Natalis on Equipollence

NATALIS ON EQUIPOLLENCE

The following is a translation (possibly the first translation into English) of part of the Summa Totius Logicae thought for centuries to have been written by Aquinas, now thought to have been written by Hervé of Nedeilec (Hervaeus Natalis), who died in 1323.

The importance of this work is its influence on later neo-scholastic logicians such as Jean Poinsot, and later still by neo-Thomists such as George Hayward Joyce. Its influence ensured that a standard view of the proposition prevailed over the views of logicians like Abelard, whose work was almost completely ignored in the scholastic tradition, and even writers such as Ockham and Buridan.

The passage here concerns 'equipollence', roughly the relation that holds between two sentences when one is true (or false) whenever the other is true (or false). While there are four standard propositions in traditional logic (A – every S is P, E – no S is P, I – some S is P and O, some S is not P), there are three signs of quantity (every, no, some) and there are two main places in which the negation sign can occur, namely before the proposition itself, and between the subject and the predicate of the proposition. Thus there are twelve possible propositions altogether, as follows

1. Omnis homo est albus (every man is white)
2. Nullus homo non est albus (no man is not white)
3. Non quidam homo non est albus (it is not the case that some man is not white)

4. Nullus homo est albus (no man is white)
5. Omnis homo non est albus (every man is not white)
6. Non quidam homo est albus (it is not the case that some man is white)

7. Quidam homo est albus (some man is white)
8. Non omnis homo non est albus (not every man is white)
9. Non nullus homo est albus (no man is white)

10. Quidam homo non est albus (some man is not white)
11. Non omnis homo est albus (not every man is white)
12. Non nullus homo non est albus (it is not the case that no man is not white)

Natalis describes how each of the non-standard propositions (2-3, 5-6, 8-9, 11-12) can be reduced to one of the standard forms 1, 4, 7 and 10, to which they are 'equipollent', i.e. one is always true, or always false, when the other is. For example

Si sumatur haec enunciatio omnis homo currit, et postponatur negatio signo universali et ponatur ad compositionem sic omnis homo non currit, quia negatio non invenit post se signum, non negat illud, et per consequens remanet universalis enunciatio. aufert autem negatio affirmationem, et sic facta est enunciatio negativa et universalis: aequipollet ergo suae contrariae, scilicet huic nullus homo currit.

If we take 'every man runs' and the negation is placed after the sign of universality and within the composition thus: 'every man does not run', because the negation happens after the sign itself, it does not negate that, and by consequence it is left as a universal assertion. But it takes away the affirmation, and so the assertion is made negative and universal, and therefore equipollent to its contrary, i.e. 'no man runs'.

Little attempt is made to justify these equivalences, beyond the rule that placing the sign of negation before the sentence gives a contradictory sentence (i.e. sentence negation), placing it after the subject gives a contrary sentence (i.e. a sentence which cannot be true at the same time as the first), and putting the sign in both places gives a subaltern (i.e. a sentence that is implied by the first. This is summarised in a little verse that is also found in Peter of Spain's Logic.

Before – contradic., after – contra.
Before and after – subalter.

These equivalences were not uncontroversial. Most medieval logicians seemed to agree that placing that 'not some' (non quidam) was equivalent to 'no' (nullus), and 'not no' to 'some', so that 3 was uncontroversially equivalent to 2, 6 to 4, 9 to 7 and 12 to 10. However, it was not universally agreed that 4 (no man is white) and 5 (every man is not white) were equivalent, nor, notoriously, 10 (some man is not white) and 11 (not every man is white). The reason is that 'some man is not white' is existential: it implies the existence of at least one man. If no men exist, it is false. By contrast 'not every man is white', or 'it is not the case that every man is white' does not seem to be existential: it merely denies that everything that is a man, is white, and can (as Abelard argued) be true when there are no men.Similarly 'every man is not white' implies the existence of at least one man, whereas 'no man is white', being equipollent to 'it is not the case that some man is white' merely denies the existence of any white man, as Abelard argued.

Two other logical works spuriously attributed to Aquinas are De Natura Syllogismorum, and De Inventione Medii.

References

Abelard, Petrus Abaelardus Dialectica, L. M. De Rijk (ed.), Assen: Van Gorcum 1970., pp. 174 ff
Grabmann, M. Beitrage Zur Geschichte der Phil. des Mittelalters, XXII, 1, 2, 1920, pp. 168-71.
Mandonnet, P. Revue Thomiste X, 1927, pp. 146-157.
Wade, F.C. Outlines of Formal Logic (transl.) 1955.

Summa totius Logicae Aristotelis, tract. 6 cap. 9

Latin English

Restat nunc dicere de aequipollentiis dictarum enunciationum. Ubi nota: quod negatio praeposita ante signum, et per consequens ante enuntiationem, aequipollet suae contradictoriae. Posita vero post signum, in compositione scilicet enunciationis, facit eam aequipollere suae contrariae. Praeposita vero et postposita facit eam aequipollere suae subalternae. Causa autem istarum aequipollentiarum est: nam in enunciationibus praedictis est considerare quantitatem, videlicet universalitatem, particularitatem: et qualitatem, scilicet negationem et affirmationem. Hoc est autem natura negationis, ut neget et tollat totum quod invenit post se. It remains now to speak of 'equipollents' of spoken assertions. Where we note, that a negation placed before the sign [of quantity], and consequently before the assertion, it is equipollent to its contradictory. But placed after the sign [of quantity, i.e. in the composition of the assertion, it makes it equipollent to its contrary. Placed both before and after, however, it makes it equipollent to its subalternate. Now the cause of these equipollencies is that in the assertions mentioned above we may consider quantity, i.e. universality and particularity, and quality, i.e. negation and affirmation. Now the nature of negation is to deny and to remove all which occurs after it.

Sic ergo ista enuntiatio omnis homo currit est universalis et affirmativa: cui si praeponatur negatio, scilicet non omnis homo currit, ista negatio tollit universalitatem, et sic remanet particularis, vel indefinita, scilicet sine signo, quae aequipollet particulari; et tollit affirmationem, et per consequens remanet negativa. Aequipollet ergo huic, scilicet quidam homo non currit, quae erat sua contradictoria. Thus the assertion 'every man runs' is universal and affirmative. If negation is placed before this, i.e. 'not every man runs', the negation removes the universality, and thus it is left as a particular, or indefinite, i.e. without sign, which is equipollent to a particular, and removes the affirmation, and in consequence is left as a negative. I.e. it is equipollent to 'a certain man does not run', which was its contradictory.

Similiter sit haec enunciatio nullus homo currit: certum est, quod haec enunciatio sit universalis et negativa: praeponatur sibi negatio, et dicatur non nullus homo currit: negatio tollit universalitatem, ergo erit particularis: tollit etiam negationem, et sic erit affirmativa, haec scilicet quidam homo currit, quae erat sua contradictoria: et idem erit de particularibus: nam haec, non quidam homo currit, aequipollet huic nullus homo currit: et propter hanc causam haec non quidam homo non currit, aequipollet huic, scilicet, quilibet homo currit. The assertion 'no man runs' is similar. It is certain that this is universal and negative: the negation is placed before it, giving 'it is not the case that no man runs'. The negation removes the universality, therefore it will be particular, and it removes also the negation, and so it will be affirmative, i.e. 'a certain man runs', which was its contradictory, and the same will be true of the particulars. For 'it is not the case that a certain man runs', is equipollent to 'no man runs', and for this reason 'it is not the case that a certain man does not run' is equipollent to 'any man whatever runs'.

Similiter etiam si sumatur haec enunciatio omnis homo currit, et postponatur negatio signo universali et ponatur ad compositionem sic omnis homo non currit, quia negatio non invenit post se signum, non negat illud, et per consequens remanet universalis enunciatio; aufert autem negatio affirmationem, et sic facta est enunciatio negativa et universalis: aequipollet ergo suae contrariae, scilicet huic nullus homo currit. Similiter haec nullus homo currit, est universalis et negativa: nam signum negativum negat compositionem enunciationis. Postponatur ergo ei negatio, et dicatur nullus homo non currit: quia negatio non habet signum post se, remanet enunciatio universalis. Et quia erat negativa quantum ad compositionem, aufert negationem, et remanet affirmativa, haec scilicet omnis homo currit, quae erat sua contraria. Similarly also, if we take 'every man runs' and the negation is placed after the sign of universality and within the composition thus: 'every man does not run', then because the negation happens after the sign itself, it does not negate that, and by consequence it is left as a universal assertion. But it takes away the affirmation, and so the assertion is made negative and universal, and therefore equipollent to its contrary, i.e. 'no man runs'. Similarly 'no man runs' is universal and negative, for the negative sign negates the composition of the assertion. Then negation is placed after it, giving 'no man does not run'. Because the negation 'not' is after the sign, it is left as a universal assertion. And because it was negative regarding the composition, it takes away the negation, and it is left as affirmative, i.e. 'every man runs', which was its contrary.

Similiter etiam sumatur ista omnis homo currit, quae est universalis et affirmativa et praeponatur sibi et postponatur negatio sic non omnis homo non currit: certum est quod secunda negatio negat ejus compositionem. Unde dato quod illa esset negativa, haec scilicet, omnis homo non currit, praeposita ergo negatio invenit post se universalitatem quam tollit, et sic facit eam particularem, invenit et negationem quam tollit, et facit eam affirmativam; et erit haec, scilicet quidam homo currit, quae erat sua subalterna. Et sic est de omnibus aliis si praeponatur et postponatur in eis negatio: quia aequipollent suo subalterno, sive sit universalis, sive particularis. Similarly also we take 'every man runs', which is universal and affirmative, and the negation is placed before and after it, thus 'it is not the case that every man does not run'. It is certain that the second negation negates the composition. Wherefore, it being given that it was negative, i.e. 'every man does not run', then the negation that is placed before it, happens after it, which takes away the universality, and thus makes it particular, and [thus] takes away the negation, and makes it affirmative. And it will be 'a certain man runs', which was its subalternate. And so it is with all others if negation is placed before and after them, because they are equipollent to their subalternate, whether universal or particular.

Notandum quod aliquando contingit quod in eadem enunciatione sunt duo signa universalia negativa: unum videlicet in subjecto, et alterum in praedicato, sicut in hac, nullus homo nullum animal est: dico quod ista aequipollet huic omnis homo aliquod animal est. Ratio est, quia quodlibet dictorum signorum et est universale, et habet in se negationem. Et quia non praeponitur negatio primo signo, remanet enunciatio universalis: ergo negatio inclusa in primo signo, quod est signum universale et negativum, praeponitur negationi, seu secundo signo, quod est signum universale, et signum negativum. Et quia invenit universalitatem, destruit ergo universalitatem, et remanet particulare: destruit negationem, et sic remanet enunciatio affirmativa, haec scilicet omnis homo aliquod animal est. Vel dicatur, sicut communiter dicitur, quod nullus non, aequipollet ei, quod est omnis, et dico non scilicet negationem quae continetur in secundo signo; non nullus vero aequipollet ei quod est quoddam, et nullus non, sumo pro negatione quae est in primo signo: et sic remanet omnis homo aliquod animal est. De omnibus his aequipollentiis, datur versus.
Prae Contradic. post Contra.
Prae postque Subalter.
Qui sic intelligitur: prae, idest negatio praeposita Contradic. idest facit aequipollere suo contradictorio: post, idest negatio postposita, facit aequipollere suo Contra., idest contrario. Prae postque idest negatio quae praeponitur et post ponitur facit aequipollere suo subalterno. Et sic patet de aequipollentiis categoricarum. It is to be noted that sometimes it happens that in the same assertion there are two universal negative signs, namely one in the subject and one in the predicate, e.g. 'no man is no animal'. I say that this is equipollent to 'every man is some animal'. The reason is, that because each of the signs above is universal, and has negation in it. And because negation is not prefixed to the first sign, it remains a universal assertion. Therefore negation is included in the first sign, which is a universal negative sign, prefixed to a negation, ie. the second sign, which is a universal negative sign. And because it finds universality, thus it destroys universality, and remains particular. It destroys negation, and thus remains an affirmative assertion, i.e. 'every man is some animal'. Or it may be said, as it commonly is, that 'no … not' is equipollent to 'every', and, I say, clearly not the negation which is contained in the second sign. But 'not no' is equipollent to 'a certain', and 'no … not' I take for the negation which is in the first sign, and thus it is left as 'every man is some animal'. About all these equipollencies, the following verse is given:
Before – contradic., after – contra.
Before and after – subalter.
Which are understood thus: 'before' i.e. the negation is put before, 'contradic.' i.e. it makes it equipollent with its contradictory. 'After', i.e. negation is placed after, and makes it equipollent with its 'contra.' i.e. its contrary. 'Before and after' i.e. negation which is placed both before and after makes it equipollent with its subalternate. And thus the equipollencies of categorical propositions is clear.

Latin	English
Restat nunc dicere de aequipollentiis dictarum enunciationum. Ubi nota: quod negatio praeposita ante signum, et per consequens ante enuntiationem, aequipollet suae contradictoriae. Posita vero post signum, in compositione scilicet enunciationis, facit eam aequipollere suae contrariae. Praeposita vero et postposita facit eam aequipollere suae subalternae. Causa autem istarum aequipollentiarum est: nam in enunciationibus praedictis est considerare quantitatem, videlicet universalitatem, particularitatem: et qualitatem, scilicet negationem et affirmationem. Hoc est autem natura negationis, ut neget et tollat totum quod invenit post se.	It remains now to speak of 'equipollents' of spoken assertions. Where we note, that a negation placed before the sign [of quantity], and consequently before the assertion, it is equipollent to its contradictory. But placed after the sign [of quantity, i.e. in the composition of the assertion, it makes it equipollent to its contrary. Placed both before and after, however, it makes it equipollent to its subalternate. Now the cause of these equipollencies is that in the assertions mentioned above we may consider quantity, i.e. universality and particularity, and quality, i.e. negation and affirmation. Now the nature of negation is to deny and to remove all which occurs after it.
Sic ergo ista enuntiatio omnis homo currit est universalis et affirmativa: cui si praeponatur negatio, scilicet non omnis homo currit, ista negatio tollit universalitatem, et sic remanet particularis, vel indefinita, scilicet sine signo, quae aequipollet particulari; et tollit affirmationem, et per consequens remanet negativa. Aequipollet ergo huic, scilicet quidam homo non currit, quae erat sua contradictoria.	Thus the assertion 'every man runs' is universal and affirmative. If negation is placed before this, i.e. 'not every man runs', the negation removes the universality, and thus it is left as a particular, or indefinite, i.e. without sign, which is equipollent to a particular, and removes the affirmation, and in consequence is left as a negative. I.e. it is equipollent to 'a certain man does not run', which was its contradictory.
Similiter sit haec enunciatio nullus homo currit: certum est, quod haec enunciatio sit universalis et negativa: praeponatur sibi negatio, et dicatur non nullus homo currit: negatio tollit universalitatem, ergo erit particularis: tollit etiam negationem, et sic erit affirmativa, haec scilicet quidam homo currit, quae erat sua contradictoria: et idem erit de particularibus: nam haec, non quidam homo currit, aequipollet huic nullus homo currit: et propter hanc causam haec non quidam homo non currit, aequipollet huic, scilicet, quilibet homo currit.	The assertion 'no man runs' is similar. It is certain that this is universal and negative: the negation is placed before it, giving 'it is not the case that no man runs'. The negation removes the universality, therefore it will be particular, and it removes also the negation, and so it will be affirmative, i.e. 'a certain man runs', which was its contradictory, and the same will be true of the particulars. For 'it is not the case that a certain man runs', is equipollent to 'no man runs', and for this reason 'it is not the case that a certain man does not run' is equipollent to 'any man whatever runs'.
Similiter etiam si sumatur haec enunciatio omnis homo currit, et postponatur negatio signo universali et ponatur ad compositionem sic omnis homo non currit, quia negatio non invenit post se signum, non negat illud, et per consequens remanet universalis enunciatio; aufert autem negatio affirmationem, et sic facta est enunciatio negativa et universalis: aequipollet ergo suae contrariae, scilicet huic nullus homo currit. Similiter haec nullus homo currit, est universalis et negativa: nam signum negativum negat compositionem enunciationis. Postponatur ergo ei negatio, et dicatur nullus homo non currit: quia negatio non habet signum post se, remanet enunciatio universalis. Et quia erat negativa quantum ad compositionem, aufert negationem, et remanet affirmativa, haec scilicet omnis homo currit, quae erat sua contraria.	Similarly also, if we take 'every man runs' and the negation is placed after the sign of universality and within the composition thus: 'every man does not run', then because the negation happens after the sign itself, it does not negate that, and by consequence it is left as a universal assertion. But it takes away the affirmation, and so the assertion is made negative and universal, and therefore equipollent to its contrary, i.e. 'no man runs'. Similarly 'no man runs' is universal and negative, for the negative sign negates the composition of the assertion. Then negation is placed after it, giving 'no man does not run'. Because the negation 'not' is after the sign, it is left as a universal assertion. And because it was negative regarding the composition, it takes away the negation, and it is left as affirmative, i.e. 'every man runs', which was its contrary.
Similiter etiam sumatur ista omnis homo currit, quae est universalis et affirmativa et praeponatur sibi et postponatur negatio sic non omnis homo non currit: certum est quod secunda negatio negat ejus compositionem. Unde dato quod illa esset negativa, haec scilicet, omnis homo non currit, praeposita ergo negatio invenit post se universalitatem quam tollit, et sic facit eam particularem, invenit et negationem quam tollit, et facit eam affirmativam; et erit haec, scilicet quidam homo currit, quae erat sua subalterna. Et sic est de omnibus aliis si praeponatur et postponatur in eis negatio: quia aequipollent suo subalterno, sive sit universalis, sive particularis.	Similarly also we take 'every man runs', which is universal and affirmative, and the negation is placed before and after it, thus 'it is not the case that every man does not run'. It is certain that the second negation negates the composition. Wherefore, it being given that it was negative, i.e. 'every man does not run', then the negation that is placed before it, happens after it, which takes away the universality, and thus makes it particular, and [thus] takes away the negation, and makes it affirmative. And it will be 'a certain man runs', which was its subalternate. And so it is with all others if negation is placed before and after them, because they are equipollent to their subalternate, whether universal or particular.
Notandum quod aliquando contingit quod in eadem enunciatione sunt duo signa universalia negativa: unum videlicet in subjecto, et alterum in praedicato, sicut in hac, nullus homo nullum animal est: dico quod ista aequipollet huic omnis homo aliquod animal est. Ratio est, quia quodlibet dictorum signorum et est universale, et habet in se negationem. Et quia non praeponitur negatio primo signo, remanet enunciatio universalis: ergo negatio inclusa in primo signo, quod est signum universale et negativum, praeponitur negationi, seu secundo signo, quod est signum universale, et signum negativum. Et quia invenit universalitatem, destruit ergo universalitatem, et remanet particulare: destruit negationem, et sic remanet enunciatio affirmativa, haec scilicet omnis homo aliquod animal est. Vel dicatur, sicut communiter dicitur, quod nullus non, aequipollet ei, quod est omnis, et dico non scilicet negationem quae continetur in secundo signo; non nullus vero aequipollet ei quod est quoddam, et nullus non, sumo pro negatione quae est in primo signo: et sic remanet omnis homo aliquod animal est. De omnibus his aequipollentiis, datur versus. Prae Contradic. post Contra. Prae postque Subalter. Qui sic intelligitur: prae, idest negatio praeposita Contradic. idest facit aequipollere suo contradictorio: post, idest negatio postposita, facit aequipollere suo Contra., idest contrario. Prae postque idest negatio quae praeponitur et post ponitur facit aequipollere suo subalterno. Et sic patet de aequipollentiis categoricarum.	It is to be noted that sometimes it happens that in the same assertion there are two universal negative signs, namely one in the subject and one in the predicate, e.g. 'no man is no animal'. I say that this is equipollent to 'every man is some animal'. The reason is, that because each of the signs above is universal, and has negation in it. And because negation is not prefixed to the first sign, it remains a universal assertion. Therefore negation is included in the first sign, which is a universal negative sign, prefixed to a negation, ie. the second sign, which is a universal negative sign. And because it finds universality, thus it destroys universality, and remains particular. It destroys negation, and thus remains an affirmative assertion, i.e. 'every man is some animal'. Or it may be said, as it commonly is, that 'no … not' is equipollent to 'every', and, I say, clearly not the negation which is contained in the second sign. But 'not no' is equipollent to 'a certain', and 'no … not' I take for the negation which is in the first sign, and thus it is left as 'every man is some animal'. About all these equipollencies, the following verse is given: Before – contradic., after – contra. Before and after – subalter. Which are understood thus: 'before' i.e. the negation is put before, 'contradic.' i.e. it makes it equipollent with its contradictory. 'After', i.e. negation is placed after, and makes it equipollent with its 'contra.' i.e. its contrary. 'Before and after' i.e. negation which is placed both before and after makes it equipollent with its subalternate. And thus the equipollencies of categorical propositions is clear.