We formalize the notion of vector semi-inner products and introduce a class seminorms which are built from these maps. The classical Pythagorean theorem parallelogram law then generalized to that have geometric mean closed lattice for codomain. In special case this codomain is square root closed, semiprime $f$-algebra, we provide sharpening triangle inequality as well condition equality.

ON THE INNER FRONT COVER: Septin‐caged Shigella flexneri in infected HEK293 cells. Bacteria and actin are labelled red, SEPT7 green, DNA blue. Credit: Gizem Ozbaykal‐Guler Serge Mostowy, London School of Hygiene & Tropical Medicine image

The contemporary discussion on the subjective character of conscious experience is characterized by a stark contrast between higher-order intentionalism, according to which any state awareness depends instantiation intentional properties mental states, and anti-intentionalism, inner constitutive fundamentally different from ordinary instances external or introspective awareness, in that one’s n...

Inner speaking is a common and widely discussed phenomenon of inner experience. Based on our studies of inner experience using Descriptive Experience Sampling (a qualitative method designed to produce high fidelity descriptions of randomly selected pristine inner experience), we advance an initial phenomenology of inner speaking. Inner speaking does occur in many, though certainly not all, mome...