Amor et odium exempli gratia erga Petrum destruitur si tristitia quam hoc et lætitia quam ille involvit, ideæ alterius causæ jungatur et eatenus uterque diminuitur quatenus imaginamur Petrum non solum fuisse alterutrius causam.
Love and hatred, for example, is destroyed towards Peter if sadness which involves the latter (hatred) or joy which involves the former (love), is joined to the idea of another cause and insofar as each of them are diminished to the extent that we imagine that Peter has not been the only cause of either.
DEMONSTRATIO: Patet ex sola amoris et odii definitione, quam vide in scholio propositionis 13 hujus. Nam propter hoc solum lætitia vocatur amor et tristitia odium erga Petrum quia scilicet Petrus hujus vel illius affectus causa esse consideratur. Hoc itaque prorsus vel ex parte sublato affectus quoque erga Petrum prorsus vel ex parte diminuitur. Q.E.D.
It is clear from the definition of love and hatred alone, which see in IIIP13S. For on account of this along joy is called love and sadness hatred toward Peter because of course Peter is considered the cause of this or that affect. Thus, this affect, removed either wholly or in part is also towards Peter diminished either wholly or in part.
In III.P48, Spinoza is accentuating our causally-oriented mind. As we register an increase or decrease in perfection, so do we follow the source. This identification process is fraught with opportunities to get it wrong and lead to false ideas of love and hatred.
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