Popper

Karl Popper asked what it means, in general, for one statement to follow from others, before fixing any particular logic. A basis answers that question by fixing a so-called ‘deducibility relation’ and a small number of closure conditions it must satisfy.

Basis I defines what a ‘deducibility relation’ is: a relation between a list of premises and a conclusion, under two closure conditions (generalized reflexivity, generalized transitivity).

Popper then develops the concepts of what is ‘demonstrable’, ‘complementary’, ‘refutable’ and ‘contradictory’, and generalizes these four concepts into one notion of ‘relative demonstrability’. Relative demonstrability takes a list of premises and a list of conclusions and rougly states that given that the premises hold, one of the conclusions must hold too.

Basis III asks for the least: a binary ‘follows’ relation (whether one statement follows from another statement), every statement follows from itself, and following composes transitively. Every Basis III generates a Basis I, so the smaller starting point loses nothing.

On top of a basis sits a second relation, demonstrability, between a list of premises and a list of conclusions. Reading several conclusions as “at least one of these holds” turns four familiar properties into the cases where one side is empty. Demonstrability also supports Cut: whenever a statement can be added to one claim’s conclusions and removed from another’s premises, chaining the two through it never needs that statement to survive.

Conjunction, disjunction, negation and implication are not built into a basis. Each is instead a property a statement can have relative to others, stated purely in demonstrability: a statement counts as a conjunction exactly when it satisfies the matching characterizing property.

Source: Binder, Piecha & Schroeder-Heister (eds.), The Logical Writings of Karl Popper, Trends in Logic 58 (2022), ISBN 978-3-030-94926-6.