| alternatives | |||||
| any no. | two | one | none | ||
| any no. | renders exh. Γ ⊨ Σ Γ, Σ⋈ ⊨ ⊥ |
entails Γ ⊨ ψ (same) |
inconsistent Γ ⊨ Γ ⊨ ⊥ |
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| two | mutually excl. φ, φ′ ⊨ φ, φ′ ⊨ ⊥ |
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| one | implies φ ⊨ ψ (same) |
absurd φ ⊨ φ ⊨ ⊥ |
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| none | exhaustive ⊨ Σ Σ⋈ ⊨ ⊥ |
jointly exh. ⊨ ψ, ψ′ ψ⋈ ⊨ ψ′ |
tautologous ⊨ ψ (same) |
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| multiple alternatives | positive conclusion |
negative: no alternatives |
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| can replace alternatives by assumptions contradictory to them |
can replace empty set of alternatives by conclusion ⊥ |
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