Formal logic. About deductive systems from axioms.

Deductive system or formal system. From axioms equipped with rules of inference come up to theorems, and so on.

Complete formal system. A deductive system is complete when all its statements (theorems, corollaries, and so on) are derivable from axioms and rules.

Inconsistent formal system. A deductive system is inconsistent when exists contradictory theorems (A and no A).

Consistent Complete formal system.

Are the "proofs by contradiction" weaker than other proofs?

Paraconsistent logic

Syllogistic logic is a kind of logic in which the fundamental elements are terms, and arguments are evaluated depending on how the terms are arranged in the argument.

Modal logic is a kind of logic that involves such concepts as a possibility, necessity, belief, and doubt.

Entails

Implies (⇒ ) vs. Entails (⊨ ) vs. Provable (⊢ )

⊨ double turnstile.

⊢ Provable

Nederpelt, Rob (2004). "Chapter 7: Strengthening and weakening". Logical Reasoning: A First Course (3rd revised ed.). King's College Publications. p. 62. ISBN 0-9543006-7-X.

Open Logic Project, First-order logic (p.7). Accessed 4 January 2022.