Exercise · ZW-CONSISTENCY-006 · ★★☆ Intermediate
A system is inconsistent if it proves...
Choose one option.
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- D. P and ¬P
Explanation
A system is inconsistent when it proves \(P\) and \(\neg P\) at once, which breaks consistency.
Review the concept: Consistency
A system \(\mathcal{S}\) is consistent if there is no formula \(\varphi\) such that \(\mathcal{S}\) proves both \(\varphi\) and \(\neg\varphi\).
\[\\neg\\exists\\varphi\\,(\\mathcal{S}\\vdash\\varphi\\wedge\\mathcal{S}\\vdash\\neg\\varphi)\]
More practice problems: consistency
- ZW-CONSISTENCY-001 A system is consistent if...
- ZW-CONSISTENCY-002 Which situation signals an inconsistency?
- ZW-CONSISTENCY-003 If a system has contradictory axioms, it may happen that...
- ZW-CONSISTENCY-004 Consistency is a property of...
- ZW-CONSISTENCY-005 To study consistency one asks whether the system can prove...
- ZW-CONSISTENCY-007 In an inconsistent classical theory the principle of... can appear
- ZW-CONSISTENCY-008 Consistency asks whether there is a simultaneous derivation of...
- ZW-CONSISTENCY-009 Which statement about consistency is correct?