Formal logic

Formal logic studies valid inference through languages with explicit syntax and semantics, and systems of rules or proofs. It abstracts argument structure from the particular subject matter of statements.

Formal language: A set of strings formed from an alphabet according to specified rules. Formal logic uses precisely specified languages to express statements and arguments.
Deductive system: A formal arrangement of axioms and inference rules for deriving conclusions. It specifies the steps by which formal proofs are constructed.
Classical logic: A family of logics governed by principles such as excluded middle and noncontradiction. It is the standard setting for many formal treatments of valid inference.
Gödel's completeness theorem: A theorem stating that first-order logical validity is equivalent to provability in a sound proof system. It establishes a precise match between semantic validity and formal proof for first-order logic.
Aristotle: An ancient Greek philosopher whose works established the first systematic study of syllogistic reasoning. His syllogistic logic is a major historical precursor to modern formal systems.
Syntax: Rules governing how symbols combine into well-formed expressions. Syntax determines which expressions count as formulas in a logical system.
Formal proof: A finite sequence of statements, each justified by an axiom or an inference rule. A proof makes an inference valid by exhibiting its permitted derivation.
Intuitionistic logic: A logic in which truth is tied to proof and the law of excluded middle is not generally accepted. It rejects an inference principle accepted in classical logic.
Gödel's incompleteness theorems: Two theorems showing limits on what sufficiently expressive, consistent formal theories can prove about arithmetic. They reveal limits on completeness and self-verification in formal systems.
Syllogistic logic: A system of deductive reasoning involving categorical propositions and their relations. It formalized patterns of inference long before modern symbolic notation.

Linked from 15 pages