The theory of existential graphs, which Peirce ultimately divided into four quadrants ((upalpha , upbeta , upgamma ) and (updelta )), is a rich method of analysis in the philosophy of logic. Its (upbeta )-part boasts a diagrammatic theory of quantification, which by 1902 Peirce had used in the logical analysis of (i) natural-language expressions such as complex donkey-type anaphora, (ii) quantificational patterns describing new mathematical concepts, and (iii) cognitive information processing. In the (upbeta )-quadrant, he came close to inventing independence-friendly logic, the idea of which he found indispensable in fulfilling the tasks (i)–(iii).