Formal Logic

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Scope

Logic is the science of correct argument. What makes a correct argument good, that is, what is it about the structure of a correct argument that guarantees that, if the premises are all true, the conclusion will be true as well?

Formal Logic is sometimes called Classical logic, for extension into non-classical logic (and new developments), see Modal Logic below and Philosophy of Logic. For Mathematical Logic including Set Theory, Model Theory, Computability and the topics now increasingly migrating into Mathematics see Philosophy of Mathematics, for challenges to first-order classical Logic see Applied Logic.

General

Metalogic: syntax and semantics of formal languages, formal systems, and logical calculi

  • The Sentential Calculus : model to show how compound sentences are built up by simpler sentences Some terminology: A sentence of the form (φ ∨ ψ) is called a disjunction, and φ and ψ are its disjuncts. A sentence of the form (φ ∧ ψ) is a conjunction, and φ and ψ are its conjuncts. ¬φ is a negation, and φ is its negatum. (φ → ψ) is a conditional, with φ as its antecedent and ψ as its consequent. (φ ↔ ψ) is a biconditional and φ and ψ are its components. Extension theorem, proof systems (method of truth tables, search for counter-examples method), soundness and completeness theorem, SC translation into English.
  • IThe propositional calculus: logic of unanalysed sentences in combination - Symbols such as not and or if.. then, is equivalent to. Propositional variables. Formation rules, validity, interdefinitability of operators. Axiomatisation. Special systems of the propositional calculus. Partial systems e.g. pure implicational calculus, three-valued or many-valued logics, intuitionistic calculus, natural deduction method. Consistency, completeness, and decidability of the propositional calculus.
  • The predicate calculus Monadic Predicate Calculus (only about properties). Full Predicate Calculus is about properties and relations, Individual variables and predicate variables, universal and existential quantifiers. The logic of individual variables. Higher order predicate calculi. Russel's theory of definite description. Boolean Logic

Interpretation of syllogistic as an axiomatic deductive system. Extension of syllogistics.

Set theory and natural-number arithmetic

The logic of classes. Arithmetic as a logical system. See Foundations of Mathematics.

Topics (for further investigation)

Causation Look at Hume's critique of causation theories.

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