Let us finally turn to nonmonotonic reasoning (for more information see the entry on non-monotonic logic). A premise [ltr]ββ[/ltr] classically entails a conclusion [ltr]γ,β⊢γγ,β⊢γ[/ltr], just in case [ltr]γγ[/ltr] is true in every model or truth value assignment in which [ltr]ββ[/ltr] is true. The classical consequence relation [ltr]⊢⊢[/ltr](conceived of as a relation between two sentences rather than as a relation between a set of sentences, the premises, and a sentence, the conclusion) is non-ampliative in the sense that the conclusion of a classically valid argument does not convey information that goes beyond the information contained in the premise.[ltr]⊢⊢[/ltr] has the following monotonicity property. For any sentences [ltr]α,β,γα,β,γ[/ltr] in [ltr]LL[/ltr]:[ltr]If α⊢γ, then α∧β⊢γ.If α⊢γ, then α∧β⊢γ.[/ltr]
That is, if [ltr]γγ[/ltr] follows from [ltr]αα[/ltr], then [ltr]γγ[/ltr] follows from any sentence [ltr]α∧βα∧β[/ltr] that is at least as logically strong as [ltr]αα[/ltr]. However, everyday reasoning often is ampliative. When Sophia sees the thermometer at 85° Fahrenheit she infers that it is not too cold to have dinner in the garden. If Sophia additionally sees that the thermometer is placed above the oven where she is boiling her pasta, she will not infer this any more. Nonmonotonic reasoning is the study of reasonable consequence relations which violate monotonicity (Gabbay 1985, Kraus & Lehmann & Magidor 1990, Makinson 1989; for an overview see Makinson 1994).For a fixed set of background beliefs [ltr]BB[/ltr], the revision operators [ltr]∗∗[/ltr] from the previous paragraphs give rise to nonmonotonic consequence relations [ltr]∣∼∣∼[/ltr] as follows (Makinson & Gärdenfors 1991):[ltr]α∣∼β if and only if β∈B∗α.α∣∼β if and only if β∈B∗α.[/ltr]
Nonmonotonic consequence relations on a language [ltr]LL[/ltr] are supposed to satisfy the following principles from Kraus & Lehmann & Magidor (1990).| (KLM1) | [ltr]α∣∼αα∣∼α[/ltr] | Reflexivity |
| (KLM2) | If [ltr]⊢α↔β⊢α↔β[/ltr] and [ltr]α∣∼γα∣∼γ[/ltr], then [ltr]β∣∼γβ∣∼γ[/ltr] | Left Logical Equivalence |
| (KLM3) | If [ltr]⊢α→β⊢α→β[/ltr] and [ltr]γ∣∼αγ∣∼α[/ltr], then [ltr]γ∣∼βγ∣∼β[/ltr] | Right Weakening |
| (KLM4) | If [ltr]α∧β∣∼γα∧β∣∼γ[/ltr] and [ltr]α∣∼βα∣∼β[/ltr], then [ltr]α∣∼γα∣∼γ[/ltr] | Cut |
| (KLM5) | If [ltr]α∣∼βα∣∼β[/ltr] and [ltr]α∣∼γα∣∼γ[/ltr], then [ltr]α∧β∣∼γα∧β∣∼γ[/ltr] | Cautious Monotonicity |
| (KLM6) | If [ltr]α∣∼γα∣∼γ[/ltr] and [ltr]β∣∼γβ∣∼γ[/ltr], then [ltr]α∨β∣∼γα∨β∣∼γ[/ltr] | Or |
The standard interpretation of a nonmonotonic consequence relation [ltr]∣∼∣∼[/ltr] is “If …, normally …”. Normality among worlds is spelt out in terms of preferential models [ltr]⟨S,l,≼⟩⟨S,l,≼⟩[/ltr] for [ltr]LL[/ltr], where [ltr]SS[/ltr] is a set of states, and [ltr]l:S→Mod[size=13]Ll:S→ModL[/ltr] is a function that assigns each state [ltr]
ss[/ltr] in [ltr]
SS[/ltr] its world [ltr]
l(s)l(s)[/ltr] in [ltr]
ModLModL[/ltr]. The abnormality relation [ltr]
≼≼[/ltr] is a strict partial order on [ltr]
ModLModL[/ltr] that satisfies a certain smoothness condition. For our purposes it suffices to note that the order among the worlds that is induced by a pointwise ranking function is such an abnormality relation. Given a preferential model [ltr]
⟨S,l,≼⟩⟨S,l,≼⟩[/ltr] we can define a nonmonotonic consequence relation [ltr]
∣∼∣∼[/ltr] as follows. Let [ltr]
α¯¯¯α¯[/ltr] be the set of states in whose worlds [ltr]
αα[/ltr] is true, i.e., [ltr]
α¯¯¯={s∈S:l(s)⊨α}α¯={s∈S:l(s)⊨α}[/ltr], and define[/size]
[ltr]α∣∼β if and only if for all s∈α[size=13]¯¯¯:(if for all t∈α¯¯¯:t⋠s, then l(s)⊨β).α∣∼β if and only if for all s∈α¯:(if for all t∈α¯:t⋠s, then l(s)⊨β).[/ltr][/size]
That is, [ltr]α∣∼βα∣∼β[/ltr] holds just in case [ltr]ββ[/ltr] is true in the least abnormal among the [ltr]αα[/ltr]-worlds. Then one can prove the following representation theorem:Theorem 2:
Let [ltr]LL[/ltr] be a language, let [ltr]B⊆LB⊆L[/ltr] be a set of sentences, and let [ltr]α∈Lα∈L[/ltr] be a sentence. Each preferential model [ltr]⟨S,l,⟩⟨S,l,⟩[/ltr] for [ltr]LL[/ltr] induces a nonmonotonic consequence relation [ltr]∣∼∣∼[/ltr] on [ltr]LL[/ltr]satisfying KLM1–6 by defining: [ltr]α∣∼βα∣∼β[/ltr] if and only if for all [ltr]s∈α[size=13]¯¯¯s∈α¯[/ltr], if for all [ltr]
t∈α¯¯¯:t⋠st∈α¯:t⋠s[/ltr], then [ltr]
l(s)⊨βl(s)⊨β[/ltr]. For each nonmonotonic consequence relation [ltr]
∣∼∣∼[/ltr] on [ltr]
LL[/ltr] satisfying KLM1–6 there is a preferential model [ltr]
⟨S,l,≼⟩⟨S,l,≼⟩[/ltr] for [ltr]
LL[/ltr] that induces [ltr]
∣∼∣∼[/ltr] in exactly this way.[/size]
Whereas the classical consequence relation preserves truth in all logically possible worlds, nonmonotonic consequence relations preserve truth in all least abnormal worlds. For a different sem 
Nonmonotonic Reasoning
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