(up to logical equivalence) that are compliant with Ï. Such an algorithm forms the basis for ... Definition 1. dnf(Ï)
Computing Compliance Ivano Ciardelli and Floris Roelofsen April 17, 2009 We specify an algorithm which computes, for a given sentence ϕ, all sentences (up to logical equivalence) that are compliant with ϕ. Such an algorithm forms the basis for practical applications of inquisitive logic. First, we specify a procedure dnf, which determines, for any formula ϕ, an equivalent formula dnf(ϕ) which is a disjunction of assertions (a disjunctive normal form). Definition 1. dnf(ϕ) is recursively defined as follows: 1. dnf(p) = p 2. dnf(⊥) = ⊥ 3. dnf(¬ψ) = ¬ψ 4. dnf(ψ ∨ χ) = dnf(ψ) ∨ dnf(χ) W 5. dnf(ψ ∧ χ) = i,j ψi ∧ χj where: • dnf(ψ) = ψ1 ∨ . . . ∨ ψn • dnf(χ) = χ1 ∨ . . . ∨ χm • i ranges over {1, . . . , n} • j ranges over {1, . . . , m} W V 6. dnf(ψ → χ) = k1 ,...,kn i ψi → χki where: • dnf(ψ) = ψ1 ∨ . . . ∨ ψn • dnf(χ) = χ1 ∨ . . . ∨ χm • i ranges over {1, . . . , n} • k1 , . . . , kn all range over {1, . . . , m} Proposition 1. For all ϕ, dnf(ϕ) is a disjunction of assertions. Proposition 2. For all ϕ, dnf(ϕ) ≡ ϕ 1
There is a close correspondence between dnf(ϕ) and the possibilities for ϕ. Proposition 3. If π is a possibility for ϕ then π is a possibility (the unique possibility) for some disjunct of dnf(ϕ). The converse, however, is not true. This is because some disjuncts of dnf(ϕ) may be entailed by others. If one disjunct α entails another β, then |α| is contained in a possibility for ϕ, but it is not identical to any such possibility. To get a full correspondence between the possibilities for ϕ and the disjuncts of dnf(ϕ), we must eliminate those disjuncts that entail others. This operation preserves logical equivalence. We call the resulting formula the clean disjunctive normal form of ϕ, cdnf(ϕ). Definition 2. cdnf(ϕ) is obtained from dnf(ϕ) by removing any disjunct that entails any other disjunct. Proposition 4. For all ϕ, cdnf(ϕ) is a disjunction of assertions. Proposition 5. For all ϕ, cdnf(ϕ) ≡ ϕ Proposition 6. π is a possibility for ϕ if and only if π is a possibility (the unique possibility) for some disjunct of cdnf(ϕ). So cdnf(ϕ) gives us, as it were, a syntactic representation of the possibilities for ϕ. This is exactly what we need to compute compliant responses. Recall the definition of compliance: Definition 3 (Compliance). ϕ is compliant with ψ, ϕ ∝ ψ, iff 1. ϕ is significant 2. every possibility in bϕc is the union of a set of possibilities in bψc 3. every possibility in bψc restricted to |ϕ| is contained in a possibility in bϕc We define an algorithm that takes a sentence ψ as its input, and yields a set comp(ψ) of sentences that are compliant responses to ψ. Definition 4 (Algorithm). 1. The algorithm takes as input a sentence ψ. If ψ is insignificant, then there are no sentences compliant with it. So comp(ψ) = ∅ in this case. 2. If ψ is significant, we first use cdnf(ψ) to compute a set of sentences that are potentially compliant with ψ. Suppose that cdnf(ψ) = ψ1 ∨ . . . ∨ ψn . First, compute the set of potentially compliant assertions, pca, as follows: pca = {!(ψi1 ∨ . . . ∨ ψim ) | i1 , . . . im ∈ {1, . . . , n}, m ≥ 1} pca consists of all formulas that are obtained from dnf(ψ) by removing some (possibly zero, but not all) disjuncts, and then taking the noninquisitive closure. These are all assertions (because of the non-inquisitive closure) whose truth-set coincides with the union of some possibilities for ψ. So all these formulas satisfy the first and second condition in the definition of compliance. 2
3. But a compliant response to ψ does not have to be an assertion. It may be an inquisitive response. For such an inquisitive response to be compliant, however, it must be equivalent with a disjunction of compliant assertions. Thus, we compute the set of potentially compliant responses, pcr, as follows: pcr = {χ1 ∨ . . . ∨ χn | 1 ≤ n ≤ |pca| and χ1 . . . χn ∈ pca} All formulas in pcr satisfy the first and the second condition in the definition of compliance, and vice versa, every formula that satisfies these two conditions is equivalent with some formula in pcr. 4. What remains to be done is to filter out those formulas in pcr that do not satisfy the third condition in the definition of compliance. To do so, we proceed as follows. Take a sentence ϕ ∈ pcr. We know that ϕ = χ1 ∨ . . . ∨ χn , where all χ’s are assertions. We have to check that every possibility for ψ, when restricted to |ϕ|, is contained in some possibility for ϕ. To do so, take a disjunct ψj of cdnf(ψ), and check if ψj ∧ !ϕ entails one of the disjuncts of ϕ. If this works for all ψj , then ϕ is compliant with ψ, otherwise it is not. Carrying out this procedure for all ϕ ∈ pcr yields the desired set of sentences comp(ψ) that are compliant with ψ. 5. Finally, there is some optional ‘cleaning up’ to do. The formulas in comp(ψ) are all disjunctions of assertions, that is, formulas in disjunctive normal form. To allow for a more intelligible output, however, we would like to bring these formulas into clean disjunctive normal form. To do so, we simply apply cdnf to every formula in comp(ψ). We claim that comp(ψ) does not just contain some sentences that are compliant with ψ, but that it contains all such sentences (up to logical equivalence). Proposition 7. ϕ is compliant with ψ iff ϕ is logically equivalent with some sentence in comp(ψ). The propositions stated here still have to be proven, but we are pretty confident that this can be done. As for the implementation, notice that the crucial steps involve checking whether one sentence entails another. Also note that the sentences involved here are always assertions. So entailment is classical in these cases, which means that we could use existing propositional entailment/satisfiability checking algorithms to carry out this task.
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