Dependencies

Axiom 5(a) for a function \(\operatorname{ob}: \mathcal P(U) \to \mathcal P(\mathcal P(U))\) says that \(\emptyset \notin \operatorname{ob}(X)\) for all \(X\in \mathcal P(U)\).

Axiom 5(b) for a function \(\operatorname{ob}: \mathcal P(U) \to \mathcal P(\mathcal P(U))\) says that

The strong version of Axiom 5(c) for a function \(\operatorname{ob}: \mathcal P(U) \to \mathcal P(\mathcal P(U))\) says that If \(Y\in \operatorname{ob}(X)\) and \(Z\in \operatorname{ob}(X)\) then \(Y\cap Z\in \operatorname{ob}(X)\).

The weak, but potentially infinite, version of Axiom 5(c) for a function \(\operatorname{ob}: \mathcal P(U) \to \mathcal P(\mathcal P(U))\) says that

Axiom 5(d) for a function \(\operatorname{ob}: \mathcal P(U) \to \mathcal P(\mathcal P(U))\) says that

Axiom 5(e) for a function \(\operatorname{ob}: \mathcal P(U) \to \mathcal P(\mathcal P(U))\) says that If \(Y\subseteq X\) and \(Z\in \operatorname{ob}(X)\) and \(Y\cap Z\ne \emptyset \), then \(Z\in \operatorname{ob}(Y)\).

The theory ADE consists of the axioms 5(a)(d)(e).

The model alive is defined by alive n X = \(\{ Y \mid X \ne \emptyset \wedge Y \supseteq X\} \).

If ob ≠ noObligations and ∀ a, ¬ quasibad ob a, then ob = alive k.

A world \(a\) is bad if ∃ (X : Finset (Fin n)), a ∈ X ∧ \(X \setminus \{ a\} \in \operatorname{ob}X\).

If bad ob a and \(X ∩ Y ∈ ob X\) then \(X ∩ Y = X\) or \(X ∩ Y = X \setminus \{ a\} \).

The theory BDE consists of the axioms 5(b)(d)(e).

The set canon\(_2\) A B X consists of all contexts that are obligatory in the context X in the model canon\(_2\) A B.

It is defined to be: if \(X \cap B = \emptyset \) then \(\emptyset \), else: if \(X \cap A = \emptyset \) then \(\{ T \mid X \cap B \subseteq T\} \), else \(\{ T \mid X \cap A \subseteq T\} \).

The set canon\(_2\)II A B X is defined by: if \(X \cap B = \emptyset \) then \(\emptyset \), else: if \(X \cap A = \emptyset \) then \(\{ Y \mid X \cap B = X \cap Y\} \) else \(\{ Y \mid X \cap A = X \cap Y\} \).

The least model of axioms 5(b), 5(d), 5(f) and the two “oughts” in 11 is canon₂ A B.

The least model \(\operatorname{ob}\) (under inclusion) of \(\mathrm{Ought}(A\mid W)\), \(\mathrm{Ought}(B\mid W \setminus A)\), and axiom 5(b) is canon₂_II A B.

If ob satisfies axioms 5(a)(b)(d)(e) then ob satisfies conditional explosion.

Conditional explosion for ob is the statement that ∀ (A B C : Finset U), A ∈ ob C → B ∩ Aᶜ ∩ C ≠ ∅ → B ∈ ob (Aᶜ ∩ C).

For all contexts \(A\), if \(A\) is obligatory in all larger contexts \(X\supseteq A\), then \(A\) is a cosubsingleton, i.e., missing at most one element (from the global context).

If \(a_1 \notin A\) and \(a_2 \notin A\) and \(a_1 \ne a_2\) then it cannot be that \(A\in \operatorname{ob}(X)\) whenever \(A\subseteq X\).

If \(A\in \operatorname{ob}(X)\) whenever \(A\subseteq X\) and \(a_1\ne a_2\) then \(A\cup \{ a_1\} \in \operatorname{ob}(A\cup \{ a_1,a_2\} )\).

Given a set \(W\) and a family \(\mathcal F\) of functions \(f :\mathscr P(W)\to \mathscr P(\mathscr P(W))\), we say that \(f_0\in \mathcal F\) is the least element of \(\mathcal F\) if for all \(f\in \mathcal F\) and all \(X\subseteq W\), \(f_0(X)\subseteq f(X)\).

The least model of a set of axioms \(\mathscr A\) concerning a variable function \(\mathrm{ob} :\mathscr P(W)\to \mathscr P(\mathscr P(W))\) is the least element of the collection \(\mathcal F\) of all functions \(\mathrm{ob} :\mathscr P(W)\to \mathscr P(\mathscr P(W))\) satisfying all axioms in \(\mathscr A\).

“Axiom” here is used to mean simply a condition on \(\mathrm{ob}\), although we may note that all the axioms 5(a)–(g) may be formulated in first-order set theory.

For all \(B\) and \(C\), if \(B\subseteq C\) is obligatory relative to \(C\) then \(C\setminus B\) is a cosubsingleton.

Suppose that for all contexts \(A\), if \(A\) is obligatory in all larger contexts \(X\supseteq A\), then \(A\) is a cosubsingleton, i.e., missing at most one element (from the global context). Then for all \(B\) and \(C\), if \(B\subseteq C\) is obligatory relative to \(C\) then \(C\setminus B\) is a cosubsingleton.

Every model of axioms 5(abcde) is either stayAlive \(a\) for some bad world \(a\), alive, or noObligations.

Let \(W\) be a finite set of possible worlds and let \(\operatorname{ob}: \mathscr P (W)\to \mathscr P(\mathscr P(W))\). The following are equivalent:

1. \(\operatorname{ob}\) satisfies the full system suggested in [ 2 ] : 5(a), (b), (c)*, (d) and (e).

2. One of the following three holds:

   1. \(\operatorname{ob}(X)=\emptyset \) for all \(X\).

   2. \(\operatorname{ob}(X)=\{ Y \mid \emptyset \ne X \subseteq Y\} \) for all \(X\).

   3. There is a distinguished possible world \(a\) (the “forbidden” world) such that for all \(X\), \(\operatorname{ob}(X)=\{ Y \mid X \cap Y \ne \emptyset \text{ and } X\setminus \{ a\} \subseteq Y\} \).

The model noObligations is defined by noObligations X = ∅.

If bad ob a and \(X \setminus \{ a\} \ne \emptyset \) then \(X ∈ \operatorname{ob}X\).

If bad ob a and X ≠ a and X ≠ ∅ then X ∈ ob X.

Assume axioms 5(abde). If \(a\) is \(\operatorname{ob}\)-bad then \(\{ a\} ∈ \operatorname{ob}\{ a\} \).

If \(X ∈ \operatorname{ob}X\) and \(Y ≠ ∅\) then \(Y ∈ ob Y \).

If \(\emptyset \ne X \setminus \{ a\} ∈ ob X\) then \(X ∈ \operatorname{ob}X\).

If \(\mathrm{univ} \setminus \{ a\} ∈ \operatorname{ob}\mathrm{univ}\) then \(\mathrm{univ} ∈ \operatorname{ob}\mathrm{univ}\).

If \(a\) is bad and \(Y \setminus \{ a\} \ne \emptyset \) then \(Y \setminus \{ a\} \in \operatorname{ob}Y\).

The world \(a\) is quasibad if ∃ (X : Finset (Fin n)) (Y : Finset (Fin n)), a ∈ X \(\setminus \) Y ∧ Y ∈ ob X.

If \(a_1\ne a_2\), \(a_1\notin A\), \(a_2\notin A\), \(\{ a_1,a_2\} \in \operatorname{ob}(\{ a_1,a_2\} )\), then it cannot be that \(A\in \operatorname{ob}(X)\) whenever \(A\subseteq X\).

If \(a_1\ne a_2\), \(a_1\notin A\), \(a_2\notin A\), \(\{ a_1,a_2\} \in \operatorname{ob}(\{ a_1,a_2\} )\), and \(A\in \operatorname{ob}(X)\) whenever \(A\subseteq X\), then \(\{ a_1\} \in \operatorname{ob}\{ a_1, a_2\} \) and \(\{ a_2\} \in \operatorname{ob}\{ a_1, a_2\} \).

The model stayAlive is defined by stayAlive e X = \(\{ Y | X \cap Y \ne \emptyset \wedge X \setminus \{ e\} \subseteq X \cap Y\} \).

If bad ob a then ob = stayAlive a.

If bad ob a then ∀ Y, stayAlive a Y ⊆ ob Y.

If bad ob a then ∀ Y, ob Y ⊆ stayAlive a Y.

If bad ob a and bad ob b then \(a = b\).