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 deducibility relation and a small number of closure conditions it must satisfy.

Basis I takes deducibility between a list of premises and a single conclusion as primitive. Basis III instead takes deducibility between two single objects as primitive, and defines the many-premise relation from it. Basis III recovers Basis I: its defined relation satisfies the same closure conditions.

On top of either basis sits a second relation, demonstrability, between a list of premises and a list of conclusions. Demonstrability supports Cut: whenever an object can be added to one derivation’s conclusions and removed from another’s premises, chaining the two through it never needs that object to survive in the final result.

Conjunction, disjunction, and classical negation are not built into a basis. Instead, each is a property an object can have relative to others, stated purely using demonstrability: an object counts as a conjunction, a disjunction, or a classical negation 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.

Module	Definitions	Abbreviations	Lemmas	Theorems	Examples	Difficulty
Basis1	2	0	2	1	0	easy
Basis3	4	0	2	2	1	moderate
Connectives	3	0	0	0	0	easy