Support conditions

The bilateral treatment for states inherits its PW partiality as follows. For any φ∈ LP hIL we say that:

φ is accepted by s (notations|=φ) iff For allw∈s,JφKw= 1

φ is rejected bys (notations = | φ) iff For allw∈s, JφKw= 0

φ is undefined in s iff For allw∈s,JφKw=N

Acceptance and Rejection Conditions

We can spell out the Acceptance (and Rejection) conditions for every operator in the logical base. Consideringα∈Atomwe have that:

M, s|=α iff For allw∈s,JαKw= 1

M, s = | α iff For allw∈s,JαKw= 0

M, s|=⊥ iff s=∅

M, s = | ⊥ always

M, s|=φψ iff M, s = | φor M, s|=ψ

M, s = | φψ iff M, s|=φandM, s = | ψ

M, s|=φ→ψ iff For allt⊆s; if For allw∈tM, w ̸ φcl _{thenM, t|=ψ}

M, s = | φ→ψ iff M, s|=φandM, s = | ψ

M, s|= Σ x.φ iff There is an∈N;M, s|=φ[n/x]

M, s = | Σ x.φ iff For alln∈N; M, s = | φ[n_/ x]

M, s|=♢φ iff For allw∈s; There is at⊆ N(w); M, t̸|=⊥andM, t|=φ

M, s = | ♢φ iff For allw∈s; For allt⊆ N(w);M, t̸|=φ

M, s|=Kaφ iff M, s|=E(a)and For allw∈s; For allt⊆σa(w);M, t̸ |= φ

M, s = | Kaφ iff M, s|=E(a)and For allw∈s;

There is at⊆σa(w)s.t. M, t̸|=⊥and M, t = | φ

Moreover since we did not treat the Knowledge operator in the previous chapter we can define its truth conditions. Ifφ∈ LP W ∪ {Kx}:

M, wKaφ iff M, wE(a)and For allw′∈σa(w)̸ φ

M, w Kaφ iff M, wE(a)and There is aw′∈σa(w);M, w′ φ

As in the previous Chapter we can introduce a “non contradiction” lemma and prove that Downward Monotonicity and World-Singleton Correspondence are respected.

Philosophical Inquisitive Logic Questioning Philosophy Facts and Lemmas

FACT 31[Downward Monotonicity] IfM, s|=φthen for allt⊆s,M, t|=φ IfM, s = | φ then for allt⊆s,M, t = | φ

Proof. In chapter B on page 95, Theorem B.2.1. FACT 32[World-Singleton Correspondence]

M,{w} |=φiffM, wφcl

M,{w} |= φiffM, w φcl

Proof. In chapter B on page 95, Theorem B.3.1.

As a corollary of the World-Singleton Correspondence we also have that and→are equivalent in Singleton states:

M,{w} |=φ→ψ iff M, wφ→ψcl

iff M,{w} |=φψ

M,{w} |= φ→ψ iff M, w φ→ψcl

iff M,{w} |= φψ Now it is possible to prove the INQ equivalent of Lemma 3.1.1:

Lemma 4.1.1. Given any model M, state s and formula φif M, s|=φ and

M, s = | φ thens=∅

Proof. We can prove this lemma by induction onφ Base case

α∈Atom Assume for some α∈ Atom, M, s |= α and M, s = | α, thus For all w∈s,JαKw= 1and For allw∈s,JαKw= 0.

Assume for contradiction that there is aw′ ∈s, thusJαKw′ = 1and

JαKw′ = 0. From Lemma 3.1.1 we know that it cannot be the case

thatJαKw′= 1 andJαKw′ = 0, thus by contradictionw′ ̸∈s.

Therefore there is now∈s, i.e. s=∅

⊥ By definition sinceM, s⊥if and only if s=∅

Inductive step (With IH being “for any φless complex than ρ, if M, s|=φ andM, s = | φthens=∅”)

Philosophical Inquisitive Logic Questioning Philosophy

ρdef=φψ Assume that for an arbitrary M& s ⊆ W, M, s |= φ ψ and

M, s = | φψ. By definition we have that either (i)[M, s|=φor

M, s = | ψ]and (ii)M, s|=φand (iii)M, s = | ψ.

If (i) is true since M, s|=φthen along with (ii) and the Inductive Hypothesiss=∅.

If (i) is true since M, s = | ψ then along with (iii) and the Inductive Hypothesiss=∅.

Thus, for every M&s⊆ W, ifM, s|=φψ and M, s = | φψ thens=∅.

ρdef=φ→ψ Assume that for an arbitrary M& s ⊆ W, M, s |= φ → ψ and

M, s = | φ→ψ. By definition we have that either (i)[For allt⊆s; if for allw∈tM, w̸ φthenM, t|=ψ]and (ii)M, s|=φand (iii)

M, s = | ψ.

By (ii), Downward Monotonicity and Singleton-World Correspon- dence we have that For all t⊆s; for allw∈t M, wφ, which by Lemma 3.1.1 implies For all t ⊆ s; for all w ∈ t M, w ̸ φ. That alongside with (i) implies that For allt⊆s,M, t|=ψand in partic- ularM, s|=ψ. That with (iii) by IH implies thats=∅.

Wlog,M&s⊆ W, ifM, s|=φ→ψandM, s = | φ→ψthens=∅. ρdef= Σ x.φ Assume that for an arbitraryM&s⊆ W,M, s|= Σ x.φandM, s = | Σ x.φ.

By definition (1) there is an∈N such thatM, s|=φ[n_/

x] and (2)

for alln′∈N,M, s = | φ[n′_/ x].

By (2) it must also be the case thatM, s = | φ[n_/

x], which along with

(1) by IH impliess=∅.

Wlog, for everyM&s⊆ W, ifM, s|= Σ x.φandM, s = | Σ x.φ then s=∅.

ρdef=♢φ Assume that for an arbitraryM&s⊆ W,M, s|=♢φandM, s = | ♢φ. Assume for Reductio that non-vacuously For allw∈sThere is at⊆

N(w)s.t. t̸=∅ andM, t|=φand For all t′ ⊆ N(w); M, t′ ̸|=φ, that is contradictory.

Thereforew̸∈sand wlogs=∅. ρdef=Kaφ Analogue to ♢φ.

Lemma 4.1.2. If M, s|=φthen For allw∈s;M, wφcl and If M, s = | φthen For allw∈s;M, w φcl

Proof. Assume M, s |= φ. By Downward Monotonicity For all {w} ⊆ s,

M,{w} |=φ, thus by World-Singleton Correspondence For all w ∈s M, w φcl. AssumeM, s = | φ. By Downward Monotonicity For all{w} ⊆s,M,{w} |= φ, thus by World-Singleton Correspondence For allw∈sM, w φcl_.

Philosophical Inquisitive Logic Questioning Philosophy Algebraic approach for Inquisitive Content

In Chapter One the inquisitive content of a proposition φ was defined as [φ]: i.e. the set of informative statessthat supportφ.

In the bilateral setting we ought once again to distinguish acceptance and rejection, therefore we say that:

In the model M, φis accepted in s (notationM, s|=φ) iffs∈[φ]a_M

In the model M, φis rejected in s (notationM, s = | φ) iffs∈[φ]r

M For a detailed characterization of the algebra for Inquisitive Content refer to section B.4 on page 100.

The Informative Content of a proposition coincides with the union over its inquisitive content, always considering separately rejection and acceptance:

FACT 33[Informative Content]

• |φcl|⊤=∪[φ]a

• |φcl_|⊥₌∪_[φ]r

Proof. In chapter B on page 95, Theorem B.6.1