Informative Content and Valuation

FACT 58[Informative Content & Valuation Correspondence] w∈ |φ|⊤ iff JφKw= 1andw∈ |φ|⊥ iff JφKw= 0

Proof. We can prove this Fact by induction: Base case

Atomic Sentences If α∈ Atom by definition of Informative Content and Valuation we have that:

w∈ |α|⊤ iffw∈ {w∈ W |JαKw= 1}iffJαKw= 1and

w∈ |α|⊥ iffw∈ {w∈ W |JαKw= 0}iffJαKw= 0

False By definition of Informative Content and Valuation we have that:

since∅=|⊥|⊤, neverw∈ ∅and neverJ⊥Kw= 1 and

sinceW=|⊥|⊥, alwaysw∈ W and alwaysJ⊥Kw= 0

Undefined By definition of Informative Content and Valua- tion we have that:

since∅=|⋆|⊤, neverw∈ ∅and neverJ⋆Kw= 1 and

since∅=|⋆|⊥, neverw∈ ∅and neverJ⋆Kw= 0

Inductive Step (With IH being “for anyφless complex thanρ, w∈ |φ|⊤ iff JφKw= 1andw∈ |φ|⊥ iff JφKw= 0 ”)

Implication: ρdef=φ→ψ

w∈ |φ→ψ|⊤ iff w∈(|φ|⊥∪ |ψ|⊤)

iff w∈ |φ|⊥ or w∈ |ψ|⊤

iff (by IH) JφKw= 0 orJψKw= 1

iff JφK∗_w= 1orJψKw= 1 iff (JφK∗_w⊔JψKw) = 1 iff (JφKw⇒JψKw) = 1 iff Jφ→ψKw= 1 w∈ |φ→ψ|⊥ iff w∈(|φ|⊤∩ |ψ|⊥) iff w∈ |φ|⊤ and w∈ |ψ|⊥

iff (by IH) JφKw= 1andJψKw= 0

iff JφKw= 1andJψK∗w= 1 iff (JφKw⊓JψK∗w) = 1 iff (JφK∗_w⊔JψK∗∗_w)∗ = 1 iff (JφK∗w⊔JψKw) = 0 iff (JφKw⇒JψKw) = 0 iff Jφ→ψKw= 0

Partial Modal Logic Questioning Philosophy Particular Quantifier: ρdef= Σ x.φ

w∈ | Σ x.φ|⊤ iff w∈ ∪ n_∈N (|φ[n_/ x]|⊤) iff There is an∈N s.t. w∈ |φ[n_/ x]|⊤

iff (by IH) there is a n∈N s.t. Jφ[n_/

x]Kw= 1 iff ⊔ n_∈N (Jφ[n_/ x]Kw) = 1 iff J Σ x.φKw= 1 w∈ | Σ x.φ|⊥ iff w∈ ∩ n∈N (|φ[n_/ x]|⊥)

iff For all n∈N, w∈ |φ[n_/ x]|⊥

iff (by IH) For all n∈N, Jφ[n_/

x]Kw= 0 iff ⊔ n∈N (Jφ[n_/ x]Kw) = 0 iff J Σ x.φKw= 0 Possible ρdef=♢φ w∈ |♢φ|⊤ iffw∈{u∈ W|(N(u)∩ |φ|⊤)̸=∅} iff (N(w)∩ |φ|⊤)̸=∅ iff there is av∈ N(w) s.t. v∈ |φ|⊤

iff (by IH) there is av∈ N(w) s.t. JφKv= 1

iff there is av∈ N(w) s.t. JφK_v= 1

iff

d

iff J♢φKw= 1

w∈ |♢φ|⊥ iffw∈{u∈ W|(N(u)∩ |φ|⊤)=∅}

iff (N(w)∩ |φ|⊤)=∅

iff For allv∈ N(w), v̸∈ |φ|⊤

iff (by IH) For allv∈ N(w),JφKw̸= 1

iff For allv∈ N(w),JφK_w= 0

iff

d

B. Philosophical Inquisitive Logic

In this section the acceptance/rejection conditions of the operators defined in section 4.1 on page 59 are listed.

B.1 Defined Logical Operators in PhIL

¬¬φdef=φ⊥ φ ψdef=¬¬(φ(¬¬ψ)) φ ψdef= (¬¬φψ) ∆

x.φdef=¬¬ Σ x.¬¬φ ⊤def=⊥⊥

(a) Defined Inquisitive Operators

¬φdef=φ→ ⊥ !φdef=¬¬φ φ∨ψdef=!(φ ψ)

φdef=¬♢¬φ φ∧ψdef=!(φ ψ) Σ x.φdef=! Σ x.φ

(b) Defined Classical Operators

?φdef=¬φ φ ∃x.φdef= Σ x.(E(x) φ) ∀x.φdef= ∆ x.(E(x)→φ) ∃x.φdef=!∃x.φ ∀x.φdef=!∀x.φ

(c) Defined hybrid Operators

M, s|=¬¬φ iff M, s = | φ M, s = | ¬¬φ iff M, s|=φ M, s|=φ ψ iff M, s|=φandM, s|=ψ M, s = | φ ψ iff M, s = | φor M, s = | ψ M, s|=φ ψ iff M, s|=φor M, s|=ψ M, s = | φ ψ iff M, s = | φand M, s = | ψ

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

M, s = | ∆ x.φ iff There is an∈N s.t. M, s = | φ[n_/ x]

M, s|=⊤ iff always

Philosophical Inquisitive Logic Questioning Philosophy

M, s|=¬φ iff For allw∈s; M, w φcl

M, s = | ¬φ iff M, s|=φ

M, s|=!φ iff For allw∈s; M, wφcl

M, s = | !φ iff For allw∈s; M, w φcl

M, s|=φ∧ψ iff For allw∈s; M, wφcl _{andM, wψ}cl

M, s = | φ∧ψ iff For allw∈s; M, w φcl _{or M, w ψ}cl

M, s|=φ∨ψ iff For allw∈s; M, wφcl orM, wψcl

M, s = | φ∨ψ iff For allw∈s; M, w φcl andM, w ψcl

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

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

M, s|= Σ x.φ iff For allw∈s; There is an∈Ns.tM, wφ[n/x]cl

M, s = | Σ x.φ iff For allw∈s; For alln∈N;M, w φ[n/x]cl

M, s|=?φ iff M, s|=φor For allw∈s;M, wφcl

M, s = | ?φ iff s=∅

M, s|=∃x.φ iff There is an∈Ns.tM, s|=E(n)andM, s|=φ[n_/ x]

M, s = | ∃x.φ iff For alln∈N;M, s = | E(n)orM, s = | φ[n/x]

M, s|=∀x.φ iff For alln∈N; For allt⊆s, if For allM, w∈t;wE(x)thenM, t|=φ[n/x]

M, s|=∀x.φ iff There is an∈Ns.t. M, s = | E(n)andM, s = | φ[n_/ x]

M, s|=∃x.φ iff For allw∈s; There is an∈Ns.tM, wE(n)and M, wφ[n/x]cl

M, s = | ∃x.φ iff For allw∈s; For alln∈N;M, w E(n)andM, w φ[n/x]cl

M, s|=∀x.φ iff For allw∈s; For alln∈N,M, w E(n)orM, wφ[n/x]cl

M, s = | ∀x.φ iff For allw∈s; There is an∈Ns.t. M, sE(n)andM, w φ[n/x]cl