Flashcards
Logic IV
Cards in this deck (24)
- 1SequentFinite set \(\Gamma\) of sentences on the lhs, the sumbol => in the middle and a finite set of sentences \(\Delta\) on the rhs
- 2DerivationAny finite sequence of sequents s.t. every sequent is either an assumption sequent or obtained from the previous sequent by the application of an inference rule
- 3D is deducible from \(\Gamma\)iff there's a derivation of \(\Gamma\) => D
- 4\(\Gamma\) is refutableiff there's a derivation of \(\Gamma\) => \(\psi\)
- 5D is deducibleiff there's a derivation of \(\psi\) => {D}
- 6A set \(\Gamma\) secures a set \(\Delta\)if every interpretation that makes all sentence in \(\Gamma\) true makes some sentence in \(\Delta\) true
- 7D is a consequence of \(\Gamma\)iff \(\Gamma\) secures {D}
- 8\(\Gamma\) is unsatisfiableiff \(\Gamma\) secures \(\psi\)
- 9D is validiff \(\psi\) secures {D}
- 10R0\(\overline{A => A}\)
- 11k\(\Gamma\) => \(\Delta\)/\(\Gamma'\) => \(\Delta'\)
- 12~R\(\Gamma\),A => A/ \(\Gamma\) => ~A,\(\Delta\)
- 13~L\(\Gamma\) => A,\(\Delta\) / \(\Gamma\),~A => \(\Delta\)
- 14vR\(\Gamma\) => A,B,\(\Delta\) / \(\Gamma\) => A v B, \(\Delta\)
- 15v L\(\Gamma\),A => \(\Delta\) \(\Gamma\),B => \(\Delta\) / \(\Gamma\),A v B => \(\Delta\)
- 16\(\exists\)L\(\Gamma\),A[s/x] =>\(\Delta\) / \(\Gamma\),\(\exists\)xA => \(\Delta\)
- 17\(\exists\)R\(\Gamma\) =>A[s/x],\(\Delta\) / \(\Gamma\) => \(\exists\)xA,\(\Delta\)
- 18R7\(\Gamma\),s=s => \(\Delta\) / \(\Gamma\) => \(\Delta\)
- 19=L1\(\Gamma\) => A(t),\(\Delta\) / \(\Gamma\), s=t => A(s),\(\Delta\)
- 20=L2\(\Gamma\),A(t) => \(\Delta\) / \(\Gamma\),s = t, A(s) =>\(\Delta\)
- 21A proof system is sound relative to semanticswhenever every proof corresponds to a valid inference
- 22A proof system is complete relative to semanticsif every valid inference corresponds to a proof
- 23Soundness TheoremEvery derivable sequent is secure
- 24Completeness TheoremEvery secure sequent is derivable
Information
- Description
- Genzen Squent Calculus, Soundness and Completness
- Created by
- kirsty.a.paton · May 2014
- Updated
- Feb 19, 2016
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