IPL.
The meanings of inquisitive semantics are sets of alternatives, where alternatives are incomparable classical meanings. This essential feature of the semantics is mirrored on the syntactic, logical level by the fact that any formula ϕ is equivalent to a disjunction of negationsdnt(ϕ).
In a way, the main result of the previous section shows that the ability to justifydnt constitutes the essence of the logicInqL. But there is even more to say: in this section we will prove thatdntis in fact a translation of InqLinto
IPL.
We will then show that the disjunctive-negative fragment of InqLcoincides with the one of IPL, and that InqL is in fact isomorphic to the disjunctive- negative fragment ofIPLthrough the translationdnt(just asCPLis isomorphic to the negative fragment of IPLthrough the translation mappingϕto¬¬ϕ). Definition 3.3.1(Translations between logics). LetL, L0 be two logics arising from entailment relations|=L and |=L0 respectively. We say that a mapping
t from formulas in the language of L to formulas in the language of L0 is a
translation from L to L0 in case for any set of formulas Θ and any formula ϕ
we have:
Θ|=Lϕ ⇐⇒ t[Θ]|=L0 t(ϕ)
wheret[Θ] ={t(θ)|θ∈Θ}.
Let us call a formuladisjunctive-negativein case it is a disjunction of negations. The following proposition says that inquisitive entailment and intuitionistic en- tailment agree as far as disjunctive-negative formulas are concerned.
Proposition 3.3.2. If ϕ is a disjunctive-negative formula and Θ a set of disjunctive-negative formulas, Θ|=InqLϕ ⇐⇒ Θ|=IPL ϕ.
Proof. Consider an arbitrary set Θ of disjunctive-negative formulas and a disjunctive- negative formulaϕ=¬ξ1 ∨ · · · ∨ ¬ξk. If Θ |=InqL ϕ, then by compactness and
the deduction theorem there must beθ1, . . . , θn ∈Θ such thatθ1 ∧ · · · ∧θn→ ϕ∈InqL.
Now since each θk is a disjunction of negations and since the distributive
laws hold in intuitionistic logic, inIPLwe can turnθ1∧· · ·∧θn into a disjunction
of conjunctions of negations. In turn, a conjunction of negations is equivalent to a negation in intuitionistic logic. So we can find formulas χ1, . . . , χm such
thatθ1 ∧ · · · ∧θn ≡IPL ¬χ1∨ · · · ∨ ¬χm. Hence,
(θ1 ∧ · · · ∧θn →ϕ)≡IPL (¬χ1 ∨ · · · ∨ ¬χm→ϕ)≡IPL
^
1≤i≤m(¬χi →ϕ)
Equivalence inIPLimplies equivalence in InqL, soθ1∧ · · · ∧θn →ϕ∈InqL
implies that V
1≤i≤m(¬χi → ϕ) ∈ InqL, which in turn means that for each
1≤i≤nwe have ¬χi →ϕ∈InqL.
Writing outϕ, this amounts to¬χi → ¬ξ1 ∨ · · · ∨ ¬ξk ∈InqL. But sinceInqL
contains the Kreisel-Putnam axiom, it follows thatW
1≤j≤k(¬χi → ¬ξj)∈InqL,
and therefore, asInqLhas the disjunction property, for some 1≤j≤kwe must have that¬χi→ ¬ξj ∈InqL⊆CPL.
Now,¬χi → ¬ξj ≡IPL ¬¬(¬χi → ¬ξj), and since CPLandIPLagree about
negations (Chagrov and Zakharyaschev, 1997, corollary 2.49), also¬χi → ¬ξj ∈
IPL, whence a fortiori ¬χi →ϕ∈IPL.
But since this can be concluded for each i, we have V
1≤i≤m(¬χi →ϕ)∈
IPL, and therefore also the equivalent formulaθ1∧ · · · ∧θn →ϕmust be inIPL.
But then obviously Θ|=IPL ϕ.
The converse implication is trivial, asInqLextendsIPL.
As a particular case of this proposition, let us remark that for any disjunctive- negative formulaϕwe have ϕ∈InqL ⇐⇒ ϕ∈IPL.
Corollary 3.3.3. dntis a translation of InqLinto IPL.
Proof. We have to show that for any Θ and anyϕ: Θ|=InqL ϕ ⇐⇒ dnt[Θ]|=IPL dnt(ϕ)
wherednt[Θ] ={dnt(θ)|θ∈Θ}. It follows from proposition 3.2.35 that Θ|=
InqLϕ ⇐⇒ dnt[Θ]|=InqLdnt(ϕ). Butdnt(ψ) is always a disjunctive-negative
formula. So, by proposition 3.3.2, dnt[Θ] |=InqLdnt(ϕ) ⇐⇒ dnt[Θ] |= IPL dnt(ϕ) and we are done.
Observe that if the maptis a translation from a logicLto another logicL0, then t naturally lifts to an embeddingt: L/≡L → L/≡L0 of the Lindenbaum-Tarski algebra of L into the Lindenbaum-Tarski algebra of L0, given by t([ψ]≡L) := [t(ψ)]≡L0.
2
2For more details on the issues of translations between logics we refer to (Epsteinet al.,
[⊥] [p] = [¬¬p] [¬p] [p∨ ¬p] = [¬p∨ ¬¬p] [>] [⊥] [p] [¬p] [p∨ ¬p] [¬¬p] [¬p∨ ¬¬p] [¬¬p→p] .. . ... [>]
Figure 3.1: Embedding of the Lindenbaum-Tarski algebra of InqL(on the left), into the Lindenbaum-Tarski algebra of IPL (the Rieger-Nishimura lattice, on the right), for the singleton set of proposition lettersP ={p}.
Since we have seen thatdntis a translation fromInqLtoIPL, the mapdnt defined by dnt([ψ]≡InqL) = [dnt(ψ)]≡IPL is an embedding of the Lindenbaum-
Tarski algebra ofInqLinto the one ofIPL. For the singleton set of propositional letters, this embedding is depicted in figure 3.1.
Now, for any ψ, dnt(ψ) is a disjunctive-negative formula. Conversely, consider a disjunctive-negative formula ψ. Since ψ ≡InqL dnt(ψ) but both
ψ and dnt(ψ) are disjunctive-negative, it follows from proposition 3.3.2 that
ψ≡IPL dnt(ψ); in other words, we have [ψ]≡IPL= [dnt(ψ)]≡IPL =dnt([ψ]≡IPL),
so [ψ]≡IPL is in the image of the embeddingdnt.
This shows that the image of the embedding dnt is precisely the set of equivalence classes of disjunctive-negative formulas. In other words, just like
CPLis isomorphic to the negative fragment ofIPL, forInqLwe have the following result.
Proposition 3.3.4. InqLis isomorphic to the disjunctive-negative fragment of
IPL.
As a corollary of the well-known fact that CPL is isomorphic to the negative fragment of IPLwe know that, for any n, there are exactly 2n+1 intuitionisti-
cally non-equivalent negative formulas inL{p1,...,pn}, just as many as there are classically non-equivalent formulas inL{p1,...,pn}.
fragment of IPLcomes with the corollary that there are exactly as many intu- itionistically non-equivalent disjunctive-negative formulas inL{p1,...,pn}as there are inquisitively non-equivalent formulas inL{p1,...,pn}.
It follows from the expressive completeness of the connectives (proposition 2.5.2) that the number of inquisitively non-equivalent formulas in L{p1,...,pn} coincides with the number of distinct inquisitive meanings built up from indices in I{p1,...,pn}, which by definition are precisely the antichains of the powerset algebra℘(I{p1,...,pn}). This algebra is isomorphic to ℘(2
n), sinceI
{p1,...,pn} =
℘({p1, . . . , pn}) contains 2n elements. Therefore, lettingD(n) denote the num-
ber of antichains of the powerset algebra℘(n), we have the following fact. Corollary 3.3.5. For any n, there are exactly D(2n) intuitionistically non-
equivalent disjunctive-negative formulas inL{p1,...,pn}.
The numbers D(n) are known as Dedekind numbers (Dedekind, 1897), and although no simple formula is known for their calculation, their values for small
nhave been computed and are available online, see for instance: www.research. att.com/~njas/sequences/A014466.
The number of inquisitive meanings in one propositional letter is 5, as dis- played by the above picture; with two letters we have 167 meanings, and with three the number leaps to 56130437228687557907787.