Dr. J.S. Lipowski (L_02.doc) 1
The relationship between provability (syntactical concept) and truthhood (semantic concept)
Connecting the syntax and the semantics of propositional logic (soundness, adequacy, completeness,consistency, and decidability)
The two independent concepts of the syntax and the semantics of the language of propositional (sentential) logic can be connected through several metatheorems.
Properties of the formal system
Soundness of the formal system
The property of the system that its all theorems correspond to the formulae which are tautologies.
( H ├ W implies H ╞ W for any set of hypotheses H and any formula W )
Soundness connects the syntax of propositional logic and the semantics of propositional logic .
It bridges the gap that exists between the fact that the system of rules of derivation is solely syntactic, and the semantic desire that the rules of derivation be truth-preserving.
Adequacy of the formal system
The property of the system that axioms and rules of inference of the system are adequate to prove all tautologies as theorems of the system. 2
Completeness of the formal system
The property of the system that all tautologies which are wffs of the system are theorems of the system, and vice versa.
Consistency of the formal system
A system of rules of derivation of propositional logic is consistent if and only if for no sentence , is it possible to derive and from the empty set of assumptions. A system of rules of derivation of propositional logic is consistent if and only if it is not the case that there is a sentence such that both ├ and ├ .
Decidability of the formal system
The property of the system that for an arbitrary formula A of the system there is an algorithm that decides (YES on NO in a finite number of step) whether A is a theorem of the system (├ A ).
Metatheorems of the formal system
Soundness metatheorem for the formal system
A system of rules of derivation of propositional logic is sound if and only if :
├ implies ╞ for any set of hypotheses and any formula
( a system of rules of derivation of propositional logic is sound if and only if whenever is derivable from , is a tautological consequence of )
Adequacy metatheorem for the formal system
(the converse of the soundness metatheorem)
╞ implies ├ for any set of hypotheses and any formula
( a system of rules of derivation of propositional logic is adequate if whenever is a semantic consequence of , is derivable from ) 3
(Godel’s) Completeness metatheorem for the formal system (the soundness metatheorem and the adequacy metatheorem combined)
├ if and only if ╞
(for any set of hypotheses and any formula ) or, equivalently, ╞ if and only if ├
(for any set of hypotheses and any formula )
( this result states the equivalence of a syntactic concept and a semantic concept )