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Maths › 03. Logic and Foundations ›

13. Formal Proof Systems

Formal proof systems study proofs as mathematical objects, with explicit syntax, fixed rules, and precisely defined notions of derivation. The chapter begins with the syntax of proofs, where a proof is treated as a finite formal object built from formulas, assumptions, axioms, and inference rules. Derivability is then introduced to express when a formula can be obtained from a given collection of assumptions inside a chosen proof system. Normal forms...

Maths › 03. Logic and Foundations ›

13.2 Derivability

13.2 Derivability Derivability is the formal relation between assumptions and conclusions. It tells us when a formula can be obtained by applying the rules of a proof system. We write: $$ \Gamma \vdash A $$ to mean that $A$ is derivable from the assumptions in $\Gamma$. Here $\Gamma$ is a collection of formulas. It may contain one formula, many formulas, or no formulas at all. For example: $$ p,\ p...