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Logic as a Tool Chapter 2: Deductive Reasoning in Propositional Logic 2.1 Logical Deductive Systems: an overview

Valentin Goranko Stockholm University October 2016

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Deductive systems

• Logical consequence:

A1, . . . , An C and logical validity: C are semantic notions, referring to the meaning of the formulae.

• Deductive systems are meant to capture logical consequence and

validity defined by the logical semantics, in terms of deductions (derivations).

• A deduction is a completely mechanical procedure, not referring to

the meaning of the occurring formulae.

• In deductive systems logical consequence is replaced by deductive

consequence and valid formulae (tautologies) – by derivable formulae (theorems).

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Basic ingredients of a deductive system

• Formal logical language. Formulae.
• Axioms. Rules of inference.
• Inference (deduction, derivation) from a set of assumptions

(premisses) in a deductive system D: A1, . . . , An ⊢D C.

• In particular, formulae derivable from no assumptions are called

theorems of D: ⊢D C.

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Soundness and completeness of a deductive system

A deductive system D is sound (correct) for a given logical semantics if D can only derive what is logically correct (valid), i.e.: A1, . . . , An ⊢D C = ⇒ A1, . . . , An C In particular: ⊢D C = ⇒ C A deductive system D is complete for a given logical semantics if D can derive every valid logical consequence, i.e.: A1, . . . , An C = ⇒ A1, . . . , An ⊢D C In particular: C = ⇒ ⊢D C A deductive system D is adequate for a given semantics if it is both sound and complete for it, i.e.: A1, . . . , An C ⇐ ⇒ A1, . . . , An ⊢D C In particular: C ⇐ ⇒ ⊢D C.

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Main types of classical deductive systems

• Axiomatic systems
• Natural deduction
• Semantic tableaux
• Sequent calculi
• Resolution