4.2.1 Entailment in PhIL
The notion of PhIL’s Positive Inquisitive Entailment (PIE), Negative Inquisitive Entailment (NIE) and Strong Inquisitive Entailment (SIE), also called Refine- ment in accordance with the definitions in the first chapter, are similar to PE, NE and SE in chapter 4 on page 59, but formulated at the state level. Weak Inquisitive Entailment has to be less direct and slightly different from the pre- vious notion, since if a state does not reject a proposition it does not imply that there is no world in the state that reject it.
Definition 27 [Strong Inquisitive Entailment]
We say thatφ refines ψ if and only if wheneverφis accepted so is ψand wheneverψ is rejected so isφ
φ
strong
|=ψiff for allMands;
Philosophical Inquisitive Logic Questioning Philosophy Definition 28 [Positive Inquisitive Entailment]
We say that φpositively INQ entailsψ if and only if wheneverφ is accepted so is ψ
φ|=ψiff for allMands; ifM, s|=φthenM, s|=ψ
Definition 29 [Negative Inquisitive Entailment]
We say thatψnegatively INQ entails φif and only if wheneverψ is rejected so is φ
ψ = | φiff for allMands; ifM, s = | ψ thenM, s = | φ
Definition 30 [Weak Inquisitive Entailment]
We say thatφWeakly INQ entails ψif and only ifφis rejected and ψ is accepted only in the inconsistent state
φ
weak
|=ψiff For allMands; ifM, s|=φandM, s = | ψthens=∅
Once again we can show that the Refinement (SIE) has many equivalent notions and that the same ordering that we had at the Possible Worlds level holds in the States level:
FACT 34[Equivalent notions of PhIL enhancement] Given φandψ the following are equivalent:
• φ
strong
|=ψ
• φ|=ψandψ = | φ
• for all M;[φ]aM⊆[ψ]Ma and[ψ]rM⊆[φ]rM
Proof. Follows from the definition and the correspondence proved in section B.5 on page 100. φ|=ψ ψ = | φ φ strong |=ψ φ strong |=ψ ψ = | φ φ strong |=ψ φ|=ψ
FACT 35[Ifφ|=ψ thenφ
weak |=ψ] φ|=ψ φ weak |=ψ
Philosophical Inquisitive Logic Questioning Philosophy Proof. Assumeφ|=ψ, by def. for allM&s⊆ W, (I) ifM, s|=φ thenM, s|=ψ.
Take an arbitraryM&s⊆ Ws.t. M, s|=φand (II)M, s = | ψ. By (I) we have thatM, s|=ψ. Thus with (II) by Lemma 4.1.1 s=∅
Wlog, for allM& s⊆ W, if M, s|=φand M, s = | ψ thens =∅, thusφ
weak
|=ψ
FACT 36[Ifψ = | φthenφ
weak |=ψ] ψ = | φ φ weak |=ψ Proof. Analogous to the previous one.
This means that with the given definitions we still have the following strength ordering: φ strong |=ψ φ|=ψ ψ = | φ φ weak |=ψ
Figure 4.2: Inquisitive Entailment relations in order of strength
4.2.2 Validity in PhIL
Similarly to the previous chapter we have that formulas are valid if the only state that can reject them is the empty set and are invalid if the only state that accept them is the empty state.
Definition 31 [Inquisitive Validity]
We say thatφis valid in PhIL if it is rejected only in the inconsistent state
|=φiff for allMands; ifM, s = | φthens=∅
Definition 32 [Inquisitive Invalidity]
We say thatψis invalid in PhIL if it is accepted only in the incon- sistent state
|
Philosophical Inquisitive Logic Questioning Philosophy Equivalence and Correspondence
Applying the definition of the Refinement and the equivalences of fact 34 on page 66 it is possible to provide many equivalent definitions of validity
FACT 37[Equivalent notions of validity and invalidity] Given φthe following are equivalent:
• |=φ • ⊤weak|=φ
• for all M;[φ]rM={∅}
Given ψthe following are equivalent: • = | ψ
• ψ
weak
|=⊥
• for all M;[ψ]aM={∅}
Proof. Follows from the definitions and the correspondence proved in section B.5 on page 100.
The Negation Switch of fact 6 on page 41 has an INQ correspondence only with the negation that preserves inquisitiveness, i.e. ¬¬φdef= (φ⊥):
FACT 38[INQ Negation Switch]
|=φiff = | ¬¬φ and = | ψ iff |=¬¬ψ
Proof.
|=φiff for allMands⊆ W; ifM, s = | φthens=∅
iff for allMands⊆ W; ifM, s = | φor M, s|=⊥thens=∅
iff for allMands⊆ W; ifM, s|=φ⊥thens=∅
iff for allMands⊆ W; ifM, s|=¬¬φthens=∅
iff = | ¬¬φ
|
= ψiff for all Mands⊆ W; ifM, s|=ψthens=∅
iff for all Mands⊆ W; ifM, s = | ¬¬ψthens=∅
iff |=¬¬ψ
Inquisitive Validity to World Validity
It is easy to show that Inquisitive Validity is preserved in Possible world validity: FACT 39[Preserved Validity]
Philosophical Inquisitive Logic Questioning Philosophy Proof.
|=φiff For allM&s; ifM, s = | φthens=∅
thus (by Mon.) For allM&s;{w} ⊆sifM,{w} |= φthen{w}=∅
thus (by Corr.) For allM&s;{w} ⊆sifM, w φcl then⊥
iff For allM&w;M, w̸ φcl iff φcl
Deduction Theorem
FACT 40[Deduction Theorem for WIE] φ weak |=ψ |=φ→ψ |=φ→ψ φ weak |=ψ Proof. φ weak
|=ψ iff For allM&s⊆ W, ifM, s|=φandM, s = | ψthens=∅
iff For allM&s⊆ W, ifM, s = | φ→ψthens=∅
iff |=φ→ψ