pp1d01 sive, seu, aut, vel

Eng.: "or". Each of these four Latin "or"-words occurs in eight clearly distinctive senses of "A or B". Moreover, these six senses are elsewhere expressed in terms other than "sive", "seu" "aut" or "vel", like "est" "idem" "idem est", "hoc est" "idem intelligo". "Sive", "seu", "aut", "vel" are linked from everywhere. Other expressions are linked from only where confusion is possible. This is by far the hardest notes page and linking procedure of this web. It is improbable that my labeling of 530 connectors is totally flawless. But the issue has to be cleared at every single locus if we ever want to understand what exactly is claimed to be proven. As everywhere, I target Spinoza's locally intended meaning, even if that seems to lead to compounded meanings that are puzzling from other perspectives. [geomap]

  Link text ‡) Meaning variant (A and B can be statements as well as concepts)
1 A [mng eqv] B Equivalent by meaning*): A or, put in another way: B
2 A [prf eqv] B Equivalent by proof: A or, as we proved to be equivalent: B. Label [prf eqv] only applied when Ethica contains a proof, else [mng eqv].
3 A [or at least, a kind of, hence] B For concepts: A is a GENUINE subset (NON-empty difference) of B.  For statements: A, NOT equivalent to B, implies B.
4 A [or even, because] B  A [because] B is equivalent to B [hence] A: A GENUINELY includes (NON-empty difference) B. For statements: A NOT equivalent, is implied by B.
5 A [excl exh] B Exclusive exhaustive: either A or B, never both, tertium non datur. (A, B) are then called a complement pair. In case of a complement triples (like "more or equal or less") the label is [excl exh triple], similarly quadruples and higher.
6 A [excl non-exh] B Exclusive non-exhaustive: either A or B, never both, but may be something else †). In all quantitative contexts, "more or less" (when used as pair, NOT as [excl exh triple] with "equal") is labeled: [excl non-exh].
7 A [non-excl exh] B Non-exclusive exhaustive: A or B or both, no other possibilities
8 A [non-excl non-exh] B Non-exclusive non-exhaustive: A or B or both, or something else, or all †)

*) we say "by meaning", not "by definition", to include, in category #1, "A or B"-strings in which the meanings of undefined terms are at stake.
†) exhaustivity is to be judged by the locally intended range of the term (exempli gratia: are we locally talking about all things, or only about extended things, about all ideas, or only about adequate ideas, etc). This is prone indeed to dissension. In all quantitative contexts, "more or less" is labled [excl non-exh], because forming an or [excl exh triple] with "equal".
 ‡) NOT linked: hanc vel illam (this or that), "unam vel ad aliquot", "two or more" when pleonastic ("...can differ...").

Explaining every single of the 530 "or"-labelling and linking decisions to this page is unfeasible. Below some examples.

{2d02 essentia}                                                                                  ... essentia, used crucially in Pars Prima, features in Pars Secunda in a formal dico (intelligo) - definition ...  the second phrasing (below vel), contrary to the first, implies symmetry: if A is part of the essence of B then B is part of the essence of A ... yet, vel is labeled [mng-eqv] because the two strings are claimed to be equivalent; readers may disagree ...
 ... I consider as belonging to the essence of a thing that, which being given, the thing is necessarily given also, and, which being removed, the thing is necessarily removed also; in other words, that without which the thing, and which itself without the thing, can neither be nor be conceived.  ...  Ad essentiam alicujus rei id pertinere dico quo dato res necessario ponitur et quo sublato res necessario tollitur; vel [mng-eqv] id sine quo res et vice versa id quod sine re nec esse nec concipi potest.
{2d08 corpora unita}                                                                        ..."same or different" is exhaustive, since there is logically no third possibility..."compelled or move...[uncompelled]...preserve...relation" is non-exhaustive: third cases are logically possible, even the usual cases. The first and second form the special cases of "corpora unita"...
...When any given bodies of the same or different magnitude are compelled by other bodies to remain in contact, or if they be moved at the same or different rates of speed, so that their mutual movements should preserve among themselves a certain fixed relation, we say that such bodies are in union, and that together they compose one body or individual, which is distinguished from other bodies by the fact of this union. ...Cum corpora aliquot ejusdem aut [excl exh] diversae magnitudinis a reliquis ita coercentur ut invicem incumbant vel [excl non-exh] si eodem aut [excl exh] diversis celeritatis gradibus moventur ut motus suos invicem certa quadam ratione communicent, illa corpora invicem unita dicemus et omnia simul unum corpus sive [mng eqv]  individuum componere quod a reliquis per hanc corporum unionem distinguitur.
{2p12 objecto ideae contingit, debet percipi}                       ...the first sive connects "id ab...percipi" to "ejus...idea", the second sive connects the entire string "Quidquid...idea" to "si...percipiatur"...
...Whatsoever comes to pass in the object of the idea, which constitutes the human mind, must be perceived by the human mind, or there will necessarily be an idea in the human mind of the said occurrence. That is, if the object of the idea constituting the human mind be a body, nothing can take place in that body without being perceived by the mind. ...Quicquid in objecto ideae humanam mentem constituentis contingit, id ab humana mente debet percipi sive [mng eqv]  ejus rei dabitur in mente necessario idea hoc est [mng eqv] si objectum ideae humanam mentem constituentis sit corpus, nihil in eo corpore poterit contingere quod a mente non percipiatur.
{2L07 motum retineat antea reliquis communicet}         ...the second sive [excl exh] has operational priority (like multiplication over addition), ranging over moveatur..quiescat...the first and the third sive [non-excl exh] form, as a pair, one single logical operator (or...or...).  The fourth, in hanc sive illam not linked  - by general protocol - note ‡) above)
....the individual thus composed preserves its nature, whether it be, as a whole, in motion or at rest, whether it be moved in this or that direction; so long as each part retains its motion, and preserves its communication with other parts as before. ...Retinet ...individuum sic compositum suam naturam sive [non-excl exh] id secundum totum moveatur sive [excl exh] quiescat sive [non-excl exh] versus hanc sive versus illam partem moveatur dummodo unaquque pars motum suum retineat eumque uti antea reliquis communicet.
{3p55c2 Nemo virtutem alicui nisi quali invidet}                 ...an ocurrence of sive [hence, a kind of]: odium is a kind of tristitia ... you are affected by odium, hence you are affected by tristitia ...
...Envy is a species of hatred ...or ...pain, that is ...a modification [Lat: affectiones] whereby a man's power of activity, or endeavour towards activity, is checked... ...Invidia est ipsum odium ...sive [hence] ... tristitia hoc est [mng eqv]  ...affectio qua hominis agendi potentia seu [mng eqv] conatus coercetur... 
 
{4p08 Cognitio boni et mali laetitiae vel tristitiae}             ...bonus^malus and laetitia^tristitia are quantitative (more^less) concepts...there is a the tertium (third logical possiblity): equal/indifferent, hence NON-exhaustive...
....knowledge of good and evil is nothing else but the emotions of pleasure or pain... PROPOSITIO VIII: Cognitio boni et mali nihil aliud est quam laetitiae vel [excl non-exh] tristitiae affectus...
 
{3post01 multis affici modis agendi potentia}                        ...postulates are ("seu") axioms...but, curiously, in this case (and in {5a02}) proven...
POSTULATES
1. The human body ... can...This postulate or axiom rests on Postulate i. and Lemmas v. and vii...
 POSTULATA
1. Corpus humanum... potest... Hoc postulatum seu [mng eqv] axioma nititur postulato 1 et lemmatibus 5 et 7...
  ...record highest "or"-density  in the substring below of {3p18}
with 20%  (10/52) of the words in one contiguous string an "or"...
...the image of a thing, regarded in itself alone, is identical, whether it be referred to time past, time future, or time present; that is ....the disposition or emotion of the body is identical, whether the image be of a thing past, future, or present. Thus the emotion of pleasure or pain is the same, whether the image be of a thing past or future.... ...rei imago in se sola considerata eadem est sive [excl exh triple]  ad tempus futurum vel [excl exh triple]  praeteritum sive [excl exh triple] ad praesens referatur hoc est ... corporis constitutio seu  [mng eqv.] affectus idem est sive [excl exh triple]  imago sit rei praeterit vel [excl exh triple] futurae sive [excl exh triple] praesentis atque adeo affectus laetitiae et tristitiae idem est sive [excl exh triple] imago sit rei praeterit aut [excl exh triple] futurae sive [excl exhtriple] praesentis...
{3p28 ad laetitiam conducere conamur promovere}           ... or even:  destruere as an extreme case of amovere ...
... remove or destroy ... ... amovere vel [or even] destruere ...
 ... in-proof-frequency record: proof of  {3p57}  with 12 "or"s...
Proof.-This proposition is evident from Ax. i. (which see after Lemma iii. Prop. xiii., Part II.). Nevertheless, we will prove it from the nature of the three primary emotions. All emotions are attributable to desire, pleasure, or pain, as their definitions above given show. But desire is each man's nature or essence (III. ix. note); therefore desire in one individual differs from desire in another individual, only in so far as the nature or essence of the one differs from the nature or essence of the other. Again, pleasure and pain are passive states or passions, whereby every man's power or endeavour to persist in his being is increased or diminished, helped or hindered (III. xi. and note). But by the endeavour to persist in its being, in so far as it is attributable to mind and body in conjunction, we mean appetite and desire (III. ix. note); therefore pleasure and pain are identical with desire or appetite, in so far as by external causes they are increased or diminished, helped or hindered, in other words, they are every man's nature; wherefore the pleasure and pain felt by one man differ from the pleasure and pain felt by another man, only in so far as the nature or essence of the one man differs from the essence of the other; consequently, any emotion of one individual only differs, &c. Q.E.D. DEMONSTRATIO: Haec propositio patet ex axiomate 1, quod vide post lemma 3 {2a3.1} scholiumque propositionis 13 partis II {non-deductive reference}. At nihilominus eandem ex trium primitivorum affectuum definitionibus demonstrabimus. Omnes affectus ad cupiditatem, laetitiam vel [excl non-exh] tristitiam referuntur ut eorum quas dedimus definitiones, ostendunt. At cupiditas est ipsa uniuscujusque natura seu [mng eqv] essentia (vide ejus definitionem in scholio propositionis 9 hujus {non-deductive reference}); ergo uniuscujusque individui cupiditas a cupiditate alterius tantum discrepat quantum natura seu [mng eqv] essentia unius ab essentia alterius differt. Laetitia deinde et tristitia passiones sunt quibus uniuscujusque potentia seu [mng eqv] conatus in suo esse perseverandi augetur vel [excl non-exh] minuitur, juvatur vel [excl non-exh] coercetur (per propositionem 11 hujus{3p11} et ejus scholium {non-deductive reference}). At per conatum in suo esse perseverandi quatenus ad mentem et corpus simul refertur, appetitum et cupiditatem intelligimus (vide scholium propositionis 9 hujus); ergo laetitia et tristitia est ipsa cupiditas sive [non-excl exh] appetitus quatenus a causis externis augetur vel [excl non-exh] minuitur, juvatur vel [excl non-exh] coercetur hoc est (per idem scholium) est ipsa cujusque natura atque adeo uniuscujusque laetitia vel [excl non-exh] tristitia a laetitia vel [excl non-exh] tristitia alterius tantum etiam discrepat quantum natura seu [mng eqv] essentia unius ab essentia alterius differt et consequenter quilibet uniuscujusque individui affectus ab affectu alterius  tantum discrepat etc. Q.E.D.
{4p28 Summum mentis bonum Dei}                                           ... seu is  [a kind of] , since there is a kind of cognoscere different from intellegere : imaginare ...
....to understand or to know... .... intelligere seu [a kind of] cognoscere...