Abstract
This paper studies the properties of eight semantic consequence relations defined from a Tarski-logic L and a preference relation ≺ . They are equivalent to Shoham’s so-called preferential entailment for smooth model structures, but avoid certain problems of the latter in non-smooth configurations. Each of the logics can be characterized in terms of what we call multi-selection semantics. After discussing this type of semantics, we focus on some concrete proposals from the literature, checking a number of meta-theoretic properties and elaborating on their intuitive motivation. As it turns out, many of their meta-properties only hold in case ≺ is transitive. To tackle this problem, we propose slight modifications of each of the systems, showing the resulting logics to behave better at the intuitive level and in metatheoretic terms, for arbitrary ≺ .
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Notes
We call a Tarski-logic any logic whose consequence relation is reflexive, monotonic and transitive. In many papers on preferential semantics, L is also assumed to be compact and supraclassical. We will make it explicit whenever we need such additional properties.
Since the publication of [9], the standard practice in papers on preferential semantics is to consider a set of states and a function f that maps this set into the set of models, where f may not be injective. For the current paper, either reading is fine: one may replace the word “model” with “state” throughout the paper and interpret the notation and definitions accordingly. We simply speak of models to avoid additional clutter.
Where X is an arbitrary set, we say that X is ≺ -smooth, or that 〈X, ≺ 〉 is smooth, iff for each x ∈ X, either x is ≺ -minimal in X, or there is a ≺ -minimal y ∈ X such that y ≺ x. Note that smoothness is different from well-foundedness, i.e. the property that there are no infinite sequences of ever “better” elements in X. Also, in the absence of transitivity or irreflexivity, even finite sets of models may not be smooth–this will be illustrated by various examples throughout the paper.
In this context, one may also refer to David Lewis’ discussion of the so-called Limit Assumption, which is equivalent to the assumption of smoothness in the current context–see [10, pp. 19-21]. In Lewis’ system of spheres, ≺ expresses similarity of worlds with respect to the actual world. He argues that in many cases, one cannot assume that there is a sphere of non-actual worlds that are “most similar” to the actual world.
See also [8, Section 5.6]: “This [ = the smoothness condition] is a difficult condition to motivate as natural [...].”
In the literature on preferences in general, this is not so–see e.g., [6, Sections 1.3 and 4.2] for an overview of criticisms on the assumption of transitivity for preference relations.
This argument is similar to one by Voorbraak [24], who considers the combination of various preference relations which are induced by incomplete and possibly conflicting pieces of information. See also Section 4.2 in [6], where it is shown that various seemingly natural ways to combine preference relations do not preserve transitivity.
Our example is structurally similar to one of Schumm [19].
See Section 2.4 for the exact definition of the transitive closure of an arbitrary relation.
See e.g., [13, p. 43].
To see why, suppose Cn X does not satisfy right absorption. Let A ∈ C n(Cn X (Γ)) − Cn X (Γ). Then by left absorption and inclusion, A ∈ Cn X (Cn X (Γ)). Hence Cn X (Cn X (Γ)) ⫅̸ Cn X (Γ).
We call a model trivial if it verifies every formula of the language. Obviously, most logics do not have trivial models; Priest’s logic LP is a notable exception.
We call Δ L-trivial iff C n L (Δ) is the set of all formulas of the language.
If one can define a logical falsum ⊥ in L, then (ii) follows from (i). Also, if the language contains a classical negation, then (ii) implies (i). However, in some logics, like e.g. Priest’s LP, this is not the case.
We do not presuppose that Γ is finite. Also, recall that the underlying logic L need not contain a classical conjunction; hence Γ′cannot always be reduced to a single formula.
In some of Schlechta’s papers, Y is called downward closed in X whenever (according to our terminology) Y is ≺ -lower in X. In [1], ≺ -lowerness is dubbed ≻ -closedness and ≺ -density is dubbed ≺ -completeness. In the same paper, Y is called ≻ -dense in X whenever (in our terms) Y is both ≺ -lower and ≺ -dense in X.
In fact, to obtain a well-defined semantic consequence relation, it suffices to have a function \(\psi ': {} \Upsilon \rightarrow \wp (\mathcal {M}(\emptyset ))\), where \(\Upsilon =_{\sf df} \{\mathcal {M}(\Gamma )\mid \Gamma \subseteq \mathcal {W}\}\). However, all selection functions from the current paper are defined generically for arbitrary sets, and hence we avoid the additional clutter that such a restriction would bring along.
In fact, it suffices that ψ obeys these conditions for all sets X, Y ∈ Υ – see also footnote 19. Again, we avoid such restrictions in this paper as it turns out that we can always guarantee the conditions wherever we need them, without restrictions on the domain.
Again, one may restrict π to the domain Υ – see footnote 19 – and still obtain a well-defined consequence relation from it. For the same reasons as before, we will not do so in this paper.
Since X ∈ Λ(X), requirement (i) is trivially fulfilled. Also, in view of the definition of ≺ -density in X, every Y ∈ Λ(X) is a subset of X, which guarantees that (ii) holds.
In the introduction of [17], Schlechta also considers the generalization of semantics in terms of a single selection function f to semantics in terms of a set F of such functions, where semantic consequence is then defined as validity in at least one set f (ℳ(Γ)) for an f ∈ F. It can be easily verified that our approach covers Schlechta’s idea as a special case. Apart from this, Schlechta’s results concern only one specific type of multi-selection semantics based on a preference relation, viz. his Limit Variant – see Section 4 below.
Where ℳ ≠ ℳ (Γ) for any Γ ⊆ 𝒲, we can simply let πns(ℳ) = {ℳ}. See also the previous footnotes concerning restrictions on the domain of π.
It was David Makinson who proposed the name singular right absorption (in personal correspondence). It is easy to check that this property does not follow from inclusion and left absorption: for a fixed letter p, let Cn X (Γ) = C n(Γ) ∪ {p} for all Γ. To see why singular right absorption and left absorption do not imply inclusion, let Cn X (Γ) = ∅ for all Γ. To see why singular right absorption and inclusion do not imply left absorption, let C n Y (Γ) = C n(Γ ∪ {q}) whenever p ∈ Γ, and C n Y (Γ) = C n(Γ) otherwise (for fixed p, q).
As with selection semantics, we may restrict these conditions to all X, Y, Z, Z 1, Z 2 ∈ Υ – see also footnote 19 – and obtain the same metatheoretic properties of the corresponding consequence relations.
To see why this holds, suppose CT holds, and let Z 1 = Z 2 = Z.
This proof and the next one are based on those for the two directions of [18, Fact 3.4.5].
As before, we let x ≼ tr y iff x ≺ tr y or x = y.
In fact, this example shows that not only cumulative transitivity fails, but also the weaker fixed point property. We leave the verification of this to the reader.
If ≺ is transitive, we can easily show that Ψ 0(X) is always ≺ -lower in X. That is, assume that (i) x ∈ Ψ 0(X), (ii) y ∈ X − Ψ 0(X) and (iii) y ≺ x. Then by (ii), there is a z ∈ min≺(X) with z ≺ y. However, by (iii) and the transitivity of ≺, also z ≺ x, and hence x ∉ Ψ 0(X), contradicting (i).
Perhaps item 3 of this fact is not that immediate. To see why it holds, suppose ≺ is transitive, and x ∈ Ψ 2(X) − Ψ 0(X). Hence (i) x is below a y ∈ Ψ 0(X), but (ii) there is a z ∈ min≺(X) such that z ≺ x. By transitivity, z ≺ y as well, and hence y ∉ Ψ 0(X) — a contradiction.
Of course, one could define ψ in an ad hoc manner, e.g. by letting \(M\in \psi (\mathcal {M}(\Gamma ))\) iff \(M\in \mathcal {M}(\Gamma )\) and M ⊧ A for all A ∈ Cn X (Γ), where X is one of the systems introduced in the next section. However, even if it would turn out to be well-behaved or even equivalent to X, the resulting system can hardly be called insightful, and we would need a multi-selection semantics to provide more insight into it. so our point here does not concern mathematical possibility, but rather mathematical elegance.
We found no proof of this lemma in the literature. In view of our proof, this can only be explained by the fact that usually, only the transitive case is considered for π0. For that case, Lemma 16 follows immediately from [18, Lemma XX, item (3)].
This result generalizes Fact 3.4.3 from [18], as Schlechta only proves cumulativity for single formulas at the right and left hand side of the turnstile and restricts the scope to the case where L = CL.
In view of Fact 10, items 5 and 6 are equivalent to items (1) and (2) from Fact 3.4.3 in [18, p. 141] in the transitive case; however, for the general case where ≺ is arbitrary, they are stronger.
For the time being one may read \(\Vdash _{\Phi ^{0}}\) as primitive. However, further on in this section, it is shown that \(\Vdash _{\Phi ^{0}}\) can be characterized in terms of a multi-selection semantics, using a function Φ 0.
This theorem could also be proved in purely set-theoretic terms, relying on Theorem 4 below. In that case, one would first prove that π0(X) ⊆ Φ 0(X) – see page 29 where the function Φ 0is defined.
To see why Φ 1(X) ⫅̸ π1(X) for every X, let X = {x, y, z} and x ≺ y ≺ z. Then the set {x, z} is strongly \(\prec ^{\mathbf {tr}}_{X}\)-dense in X, but not \(\prec ^{\mathbf {tr}}_{X}\)-lower in X.
This point refutes an earlier claim by Makinson – see [13, p. 74].
Note that the requirement that π0(X) is ≺ -dense in X forces us to select some non-minimal models in cases where \(\langle \mathcal {M}(\Gamma ),\prec \rangle \) is not smooth.
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We are greatly indebted to David Makinson and the anonymous referees for helpful comments and suggestions.
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Van De Putte, F., Straßer, C. Preferential Semantics using Non-smooth Preference Relations. J Philos Logic 43, 903–942 (2014). https://doi.org/10.1007/s10992-013-9302-6
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DOI: https://doi.org/10.1007/s10992-013-9302-6