Petrus Hispanus

This article deals with the logician Petrus Hispanus; for the physician and later Pope of the same name, possibly identical to this, see Johannes XXI.

Petrus Hispanus is an important logician of the 13th century. He wrote twelve tracts around 1240 , which were later handed down under the title " Summulae logicales ". They represent the most popular medieval introduction to logic with a long history of impact.

Authorship

The logician Petrus Hispanus is traditionally identified with the Portuguese physician Petrus Hispanus (1205–1277), who became Pope John XXI in the last year of his life . was appointed. But this is not certain. Various Dominicans are also discussed as alternative authors of the Summulae logicales . There is no Greek model by Michael Psellos ; a later back translation of the treatises of Petrus Hispanus into Greek was slipped on him. The syllogistic according to Petrus Hispanus is largely identical to that of William of Sherwood ; the dating of their writings is estimated differently, so that the priority cannot be clearly determined. The memorable depiction of Aristotelian logic for scholastic teaching only achieved popularity through Petrus Hispanus. Already in Dante's Divine Comedy he was praised among the wisdom teachers in the sunny sky of Paradiso as Pietro Ispano lo qual già luce in dodici libelli (Petrus Hispanus, whose light already shines in the twelve little books). His Summulae logicales was published and commented on again and again and was widely used in universities until the 17th century. The coding of the Aristotelian syllogistics contained therein is still used today.

Mnemonic syllogistics

In the fourth treatise, Petrus Hispanus reported on the assertoric syllogistics of Aristotle and added a mnemonic technique . He translated the Aristotelian sentences into an understandable language and shortened them symbolically. The translation does not affect the logical content of the original syllogistics. Therefore, the only logical progress is the coding, which comes close to a calculus and has a mnemonic purpose. The latter is concentrated in a memorable poem that lists 19 Aristotelian syllogisms and names them:

Barbara Celarent Darii Ferio Baralipton Celantes Dabitis Fapesmo Frisesomorum Cesare Cambestres Festino Barocho Darapti Felapto Disamis Datisi Bocardo Ferison

Coding the statements

Petrus Hispanus preferred the older, more understandable categorical statements with interchanged terms to the original statements from analytics and abbreviated them using vowel codes so that they can easily be translated into formulas:

code	Names	categorical statements		Formulas	Statements of the analytics
a	universally affirmative	omnis A est B	Every A is a B.	AaB	B belongs to every A.
e	universally negative	nullus A est B	No A is a B.	AeB	B does not belong to any A.
i	particular affirmative	quidam A est B	Any A is a B.	AiB	B belongs to some A.
O	particulate negative	quidam A non est B	Any A is not a B.	AoB	B does not belong to any A.

Coding of the syllogisms

The Aristotelian syllogisms are noted here schematically: premise 1, premise 2 → conclusion. Petrus Hispanus named the first premise major and the second minor and wrote them vertically one above the other. Aristotle divided the syllogisms into three figures, which differ in the position of the upper term A in premise 1, the middle term B in both premises, and the subterm C in premise 2. Aristotle submitted syllogisms with converted conclusions, but did not include them in the first figure, as did Petrus Hispanus (Table, Figure 1a). Since he also swapped the terms, his figures look different than in the original: There AaB would be the original statement "A comes to every B" and the first syllogism would be the transit law AaB, BaC → AaC; this original form disappears in the representation with interchanged terms. So all syllogisms are outwardly rewritten. The first three vowels of their memorized names each name the statements that occur in sequence (in bold in the table).

figure	syllogism	Nickname	Example of Peter Hispanus
Figure 1 BxA, CyB → CzA	BaA, CaB → CaA	B a rb a r a	Every living being is a being Every person is a living being So: every person is a being
	BeA, CaB → CeA	C e l a r e nt	No living being is a stone Every person is a living being So: No person is a stone
	BaA, CiB → CiA	D a r ii	Every living being is a being. Some human is a living being. So: some human is a being
	BeA, CiB → CoA	F e r io	No living being is a stone. Any human being is a living being. So: any human is not a stone
Figure 1a BxA, CyB → AzC	BaA, CaB → AiC	B a r a l i pton	Every living being is a being Every person is a living being So: Any being is a human being
	BeA, CaB → AeC	C e l a nt e s	No living being is a stone Everyone is a living being So: no stone is a person
	BaA, CiB → AiC	D a b i t i s	Every living being is a being. Some human is a living being. So: some being is a human
	BaA, CeB → AoC	F a p e sm o	Every living being is a being No stone is a living being So: Any being is not a stone
	BiA, CeB → AoC	Fr i s e s o morum	Any living being is a being No stone is a living being So: any being is not a stone
Figure 2 AxB, CyB → CzA	AeB, CaB → CeA	C e s a r e	No stone is a living being Every person is a living being So: No one is a stone
	AaB, CeB → CeA	C a mb e str e s	Every person is a living being No stone is a living being So: No stone is a person
	AeB, CiB → CoA	F e st i n o	No stone is a living being. Any human is a living being. So: any human is not a stone
	AaB, CoB → CoA	B a r o ch o	Every human being is a living being. Some stone is not a living being. So: some stone is not human
Figure 3 BxA, ByC → CzA	BaA, BaC → CiA	D a r a pt i	Every person is a being Every person is a living being So: Any living being is a being
	BeA, BaC → CoA	F e l a pt o	Nobody is a stone Every person is a living being So: any living being is not a stone.
	BiA, BaC → CiA	D i s a m i s	Any person is a being Every person is a living being So: Any living being is a being
	BaA, BiC → CiA	D a t i s i	Every person is a being Every person is a living being So: Any living being is a being
	BoA, BaC → CoA	B o c a rd o	Any person is not a stone Everyone is a living being So: Any living being is not a stone
	BeA, BiC → CoA	F e r i s o n	No human is a stone Any human is a living being So: Any living being is not a stone

Coding the arguments

The arguments that Aristotle used in his proofs were abbreviated by Petrus Hispanus with consonants, each with an initial of a typical word in the argument name. In this way he completely coded the Aristotelian axiom system of syllogistics according to the following table:

code	Argument name	Aristotelian rule formalized
B.	B arbara	BaA, CaB → CaA	perfect syllogisms (axioms)
C.	C elarent	BeA, CaB → CeA
D.	D arii	BaA, CiB → CiA
F.	F erio	BeA, CiB → CoA
s	conversio s implex	AeB → BeA AiB → BiA	Conversions
p	conversio p er accidens	AaB → BiA
m	transpositio in premissis de m ajori minorem	A, B → B, A	Exchange of premises
c	per impossibile ex opposito c onclusionis	A, ¬C → contradiction equivalent to A → C	indirect proof by negating o and i
	equipollet suo c ontradictorio	¬ (AoB) = AaB ¬ (AiB) = AeB

Coding the Evidence

The notation names encode the syllogisms including evidence. Petrus Hispanus made sure that only those code consonants appear in memorized names for which the associated rule is also applicable; hence perfect syllogisms as axioms have no code consonants other than their initial. The following table highlights the code consonants in bold and transfers the coding into the Aristotelian proofs, which are clearly and precisely understandable:

syllogism	Evidence Code	proof
BaA, CaB → CaA	B arbara	Axiom not to prove
BeA, CaB → CeA	C elarent	Axiom not to prove
BaA, CiB → CiA	D arii	Axiom not to prove
BeA, CiB → CoA	F erio	Axiom not to prove
BaA, CaB → AiC	B arali p ton	BaA, CaB B arbara CaA conversio p er accidens AiC
BeA, CaB → AeC	C elante s	BeA, CaB C elarent CeA conversio s implex AeC
BaA, CiB → AiC	D abiti s	BaA, CiB D arii CiA conversio s implex AiC
BaA, CeB → AoC	F a p e sm o	BaA, CeB conversio p er accidens AiB, CeB conversio s implex AiB, BeC de m ajori minorem BeC, AiB F erio AoC
BiA, CeB → AoC	F ri s e s o m orum	BiA, CeB conversio s implex AiB, CeB conversio s implex AiB, BeC de m ajori minorem BeC, AiB F erio AoC
AeB, CaB → CeA	C e s are	AeB, CaB conversio s implex BeA, CaB C elarent CeA
AaB, CeB → CeA	C a m be s tre s	AaB, CeB de m ajori minorem CeB, AaB conversio s implex BeC, AaB C elarent AeC conversio s implex CeA
AeB, CiB → CoA	F e s tino	AeB, CiB conversio s implex BeA, CiB F erio CoA
AaB, CoB → CoA	B aro c o	ex opposito c onclusionis AaB, CaA, CoB B arbara CaB, CoB impossibilis (contradiction)
BaA, BaC → CiA	D ara p ti	BaA, BaC conversio p er accidens BaA, CiB D arii CiA
BeA, BaC → CoA	F ela p to	BeA, BaC conversio p er accidens BeA, CiB F erio CoA
BiA, BaC → CiA	D i s a m i s	BiA, BaC conversio s implex AiB, BaC de m ajori minorem BaC, AiB D arii AiC conversio s implex CiA
BaA, BiC → CiA	D ati s i	BaA, BiC conversio s implex BaA, CiB D arii CiA
BoA, BaC → CoA	B o c ardo	ex opposito c onclusionis BoA, CaA, BaC B arbara BoA, BaA impossibilis (contradiction)
BeA, BiC → CoA	F eri s on	BeA, BiC conversio s implex BeA, CiB F erio CoA

Code variants

The memory poem is circulating today in different versions. The core of Figures 1–3 remained unchanged except for the orthographic variants: Camestres, Felapton, Baroco. The inserted Figure 1a was later replaced by Figure 4, which only interchanges the premises of the syllogisms and renames the variables in order to achieve the other conclusion form CzA. The proofs then run analogously, but required new notation names that insert or delete the code m; various art names have been in use since the 17th century:

Figure 4	syllogism	Nickname	English tradition
AxB, ByC → CzA	AaB, BaC → CiA	B a m ali p	B ra m anti p
	AaB, BeC → CeA	C ale m e s	C a m ene s
	AiB, BaC → CiA	D i m ati s	D i m ari s
	AeB, BaC → CoA	F e s a p o	F e s a p o
	AeB, BiC → CoA	F re s i s on	F re s i s on

Followers of Aristotle completed the list of 19 Aristotelian syllogisms to include all 24 possible syllogisms. In Aristotle they added missing subalternations of the syllogisms Barbara, Celarent, Camestres, Cesare, Calemes, which have been referred to with modified names since the 16th century, but which do not encode the evidence by subalternation (ps or cps):

figure	syllogism	Nickname	Proof code
Figure 1	BaA, CaB → CiA	Barbari	Barbara ps
	BeA, CaB → CoA	Celaront	Celarent cps
Figure 2	AeB, CaB → CoA	Cesaro	Cesare cps
	AaB, CeB → CoA	Camestros	Cambestres cps
Figure 4	AaB, BeC → CoA	Calemos	Calemes cps

Reduced syllogistics

Petrus Hispanus coded only a small part of the logic of Aristotle. He excluded the complicated and controversial modal syllogistics. His code only captures the convincing core of assertoric syllogistics, but by no means everything from it. For example, he ignored all the falsifications with which Aristotle demonstrated using examples that there are no syllogisms with other premises. He also did not code the indirect proofs of Darii and Ferio of Figure 1, which Aristotle later submitted in order to reduce his system of axioms, nor did his indirect proof of the second conversion.

Reduction of the axiom system
BaA, CiB → CiA	Darii	ex opposito c onclusionis BaA, CeA, CiB Cambestres CeB, CiB contradiction
BeA, CiB → CoA	Ferio	ex opposito c onclusionis BeA, CaA, CiB Cesare CeB, CiB contradiction
AiB → BiA	conversio simplex 2	ex opposito c onclusionis AiB, BeA conversio s implex 1 AiB, AeB contradiction

Even so, Petrus Hispanus achieved sustained success with his coded syllogistics. The derivation of his system was also based on George Boole's mathematical logic with definitions that Leibniz had given 160 years earlier, but not published:

Definitions in Boolean Algebra
universal statements	XaY: = X¬Y = 0	XeY: = XY = 0
particular statements	XoY: = X¬Y ≠ 0	XiY: = XY ≠ 0
linked statements	A, B: = AB	A → C: = A = AC

With these definitions, Boole proved the coded rules assuming non-empty terms. This is only necessary for the conversion p and the syllogisms that have been proven with it. If one does not want to forbid empty terms, one must assume non-empty terms in these cases:

Theorem Variants in Boolean Algebra
AaB, A ≠ 0 → BiA	p conversio per accidens	BaA, CaB, C ≠ 0 → CiA	Barbari
BaA, CaB, C ≠ 0 → AiC	Baralipton	BeA, CaB, C ≠ 0 → CoA	Celaront
BaA, CeB, B ≠ 0 → AoC	Fapesmo	AeB, CaB, C ≠ 0 → CoA	Cesaro
BaA, BaC, B ≠ 0 → CiA	Darapti	AaB, CeB, C ≠ 0 → CoA	Camestros
BeA, BaC, B ≠ 0 → CoA	Felapto	AaB, BeC, C ≠ 0 → CoA	Calemos

With slightly modified definitions XaY: = (X¬Y = 0) (X ≠ 0) and XoY: = ¬ (XaY), however, the syllogistics of Aristotle results exactly. So Petrus Hispanus was already translating it into a fairly perfect consistent calculus; In addition, he created his examples consistently in a well-defined model: In an eight-valued Boolean algebra with equality, HUMAN and STONE are set as minimal non-empty terms and, in addition, LIVING BEING = NOT STONE and BEING = 1. This gives the smallest model in which these terms are different and the statements of the syllogism examples are all true. One can also understand all Aristotelian falsifications in this model.

Porphyry tree

In Tractatus II, Chapter 11 of the Summulae logicales, Petrus Hispanus coined the term porphyry tree as the name for the tree that visualized the classification system of Porphyry .

Works

• Petrus Hispanus: Tractatus = Summulae logicales , ed.LM De Rijk, Assen, 1972.

German translation: Petrus Hispanus: Logical treatises . From the Latin by W. Degen and B. Bapst, Munich 2006, ISBN 3-88405-005-2 .

Web links

• Summulae logicales with commentary by Georgius Bruxellensis, Lyon 1497
• Summulae logicales with gloss by Johannes de Magistris, Lyon 1498
• Summulae logicales with commentary by Petrus Tataretus, 1500
• Summulae logicales, annotated edition 1572 (easy to read)
• Joke Spruyt: Peter of Spain (2001), in: Stanford Encyclopedia of Philosophy (English)

Individual evidence

1. ↑ Traditional attribution up to date: W. Degen and B Bapst: Logische Abhandlungen , Munich 2006, foreword.
2. ^ Ángel d'Ors: Petrus Hispanus OP, Auctor Summularum (I). In: Vivarium . 35, 1 (1997), pp. 21-71. Ángel d'Ors: Petrus Hispanus OP, Auctor Summularum (II): Further documents and problems. In: Vivarium. 39.2 (2001), pp. 209-254. Ángel d'Ors: Petrus Hispanus OP, Auctor Summularum (III). "Petrus Alfonsi" or "Petrus Ferrandi"? In: Vivarium. 41,2 (2004), pp. 249-303.
3. ↑ In addition the well-founded bibliography: Paul Moore: Iter Psellianum. Toronto 2005, MISC 59.
4. ^ ^{A b} William of Sherwood: Introductiones in logicam III. He does not code the evidence correctly: indirect evidence through B r, which does not match Barbara and Baralipton, and the code word Campestres (= fields) with code p too much (hence Petrus Hispanus Cambestres and the later tradition Camestres as a meaningless word).
5. ^ Dante: Divina Comedia , Paradiso XII, 134f. German online: [1]
6. ↑ ^{a b c d} Petrus Hispanus, Summulae logicales , Tractatus IV 13, memorial poem with original orthography, there in large letters.
7. ^ Translations according to: Aristoteles: Topik II 1, 108b35ff, Aristoteles: De interpretatione 7, 17b17-212
8. ↑ Aristotle: An.pr. ( first analysis ) A1, 24a18f
9. ↑ ^{a b} Aristotle: An.pr. A7 29a24-27
10. ↑ ^{a b} Petrus Hispanus, Summulae logicales , Tractatus IV 6, IV 8f, IV 11, each verbally described syllogism, example, evidence sketch with arguments (determined from An.pr. A4-7).
11. ↑ Aristotle: An.pr. A4 25b37b-26a2, 26a23-28, perfect syllogisms (axioms)
12. ↑ Aristotle: An.pr. A5 27a5-39
13. ↑ Aristotle: An.pr. A6 28a17-35
14. ↑ Aristotle: An.pr. A4, 25b32ff.
15. ↑ Aristotle: An.pr. A2, 25a15-22.
16. ↑ Seldom explicitly mentioned, for example: Aristoteles: An.pr. B4, 57a17 μετάθεσις.
17. ↑ Summule logicales IV. 9
18. ↑ Aristotle: An.pr. B14, 62b29-35.
19. ↑ Summule logicales I 12, I eighteenth
20. ↑ Apuleius: Peri Hermeneias. In: C. Moreschini (ed.): De Philosophia libri. Stuttgart / Leipzig 1991, pp. 189–215, p. 213 refers to three primary and two secondary subalternations of Ariston of Alexandria, a peripatetic of 1./2. Century whose writings are lost.
21. ↑ The oldest source is likely to be: Alexander Achillini: De potestate syllogismis , Edition 1545, p. 155 [2]
22. ↑ An.pr. A8-22 (14 chapters).
23. ↑ Aristotle: An.pr. A2, 25a20f indirect proof of the 2nd conversio simplex. An.pr. A7, 29b9-14 proof from Darii and Ferio.
24. ^ George Boole: The mathematical Analysis of Logic , 1847; P. 31 mnemonic verses (English tradition) [3] ; S. 20f Definitions: ¬x: = 1-x, a as x (1-y) = 0, e as xy = 0, i as v = xy, o as v = x (1-y) with variables for element-containing Classes according to p. 15 (v can be eliminated with v ≠ 0). Linked statements: p. 51 conjunction as xy, p. 54 (36) implication x (1-y) = 0 with reference to p. 21 (4) with the equivalent formula xy = x (table).
25. ↑ Leibniz: Generales Inquisitiones , 1686, edited 1903: §151 categorical statements with 'est res' for ≠ 0 and 'non est res' for = 0; §198,6 sets the implication synonymously to 'A continet B', which §16 / §83 defines as A = AB.
26. ^ George Boole: The mathematical Analysis of Logic : p. 15 element-containing classes; P. 26ff simple conversion (s), conversion per accidens (p); P. 34 Barbara (B), Celarent (C), Exchange of Premises (m); the equivalence of the implication formulas (previous footnote) is the indirect proof (c).