Conditional Obligations in Justification Logic

Abstract

This paper presents a justification counterpart for dyadic deontic logic, which is often argued to be better than Standard Deontic Logic at representing conditional and contrary-to-duty obligations, such as those exemplified by the notorious Chisholm’s puzzle. We consider the alethic-deontic system (E) and present the explicit version of this system (JE) by replacing the alethic Box-modality with proof terms and the dyadic deontic Circ-modality with justification terms. The explicit representation of strong factual detachment (SFD) is given and finally soundness and completeness of the system (JE) with respect to basic models and preference models is established.

F. Faroldi was supported by the Ambizione grant PZ00P1$\_$201906, A. Rohani and T. Studer were supported by the Swiss National Science Foundation grant 200020$\_$184625.

A Soundness and Completeness with Respect to Basic Models

Theorem 4

System $\textsf{JE}_{\textsf{CS}}$ is sound with respect to the class of all basic models.

Proof

The proof is by induction on the length of derivations in $\textsf{JE}_{\textsf{CS}}$. For an arbitrary basic model $\varepsilon$, soundness of the propositional axioms is trivial and soundness of $\textsf{S5}$ axioms $\textsf{j}, \textsf{jt}, \textsf{j4}, \textsf{j5}, \mathsf {j+}$ immediately follows from the definition of basic evaluation and factivity. We just check the cases for the axioms containing justification terms. Suppose $\textsf{JE}_{\textsf{CS}}\vdash F$ and F is an instance of:

• (COK): Suppose $\varepsilon \Vdash [t] (B \rightarrow C/A)$ and $\varepsilon \Vdash [s] (B/A)$. Thus we have

  $$ (B \rightarrow C ,A) \in \varepsilon (t) \quad \text {and}\quad (B,A) \in \varepsilon (s). $$

  By the definition of basic model, we have $\varepsilon (t) \ominus \varepsilon (s) \subseteq \varepsilon (t \cdot s)$ and as a result $(C,A) \in \varepsilon (t \cdot s)$, which means $\varepsilon \Vdash [t \cdot s](C/A)$.

• (Nec): Suppose $\varepsilon \Vdash (\lambda : A)$. Thus $A \in \varepsilon (\lambda )$. By the definition of $\textsf{n}(\varepsilon (\lambda ))$ we have $(A,B) \in \textsf{n}(\varepsilon (\lambda ))$ for any $B \in \textsf{Fm}$ and by the definition of basic evaluation $\textsf{n}(\varepsilon (\lambda )) \subseteq \varepsilon (\textsf{n}(\lambda ))$, so $(A,B) \in \varepsilon (\textsf{n}(\lambda ))$, which means $\varepsilon \Vdash [\textsf{n}(\lambda )] (A/B)$.

• (Ext): Suppose $\varepsilon \Vdash \lambda : (A \leftrightarrow B)$, so $(A \leftrightarrow B) \in \varepsilon (\lambda )$. Since $\varepsilon (\lambda ) \odot \varepsilon (t) \subseteq \varepsilon (\textsf{e}(t, \lambda ))$, we have $(C,B) \in \varepsilon (\textsf{e}(t, \lambda ))$ if $(C,A) \in \varepsilon (t)$. Hence $\varepsilon \Vdash ( [t] (C/A) \rightarrow [\textsf{e}(t, \lambda )] (C/B))$.

• (Sh): Suppose $\varepsilon \Vdash [t] (C/A \wedge B)$, then $(C,(A \wedge B)) \in \varepsilon (t)$. By definition of $\nabla (\varepsilon (t))$ we have $(B \rightarrow C , A) \in \nabla (\varepsilon (t))$ and by definition of basic models, $\nabla \varepsilon (t) \subseteq \varepsilon (\nabla t)$. As a result, $((B \rightarrow C),A) \in \varepsilon (\nabla t)$ which means $\varepsilon \Vdash [\nabla t](B \rightarrow C/A)$.

For the axioms (Abs) and (Id) soundness is immediate from the definition of basic evaluation.    $\square$

Theorem 5

System $\textsf{JE}_{\textsf{CS}}$ is complete with respect to the class of all basic models.

Proof

Given a maximal consistent $\varGamma$, we define the canonical model $\varepsilon ^c$ induced by $\varGamma$ as follows:

• $\varepsilon ^c _{\varGamma } (P):=1$, if $P \in \varGamma$ and $\varepsilon ^c :=0$, if $P \notin \varGamma$;

• $\varepsilon ^c _{\varGamma } (\lambda ) :=\{ F \ | \ \lambda :F \in \varGamma \}$;

• $\varepsilon ^c _{\varGamma } (t) :=\{ (F,G) \ | \ [t](F/G) \in \varGamma \}$.

We first show that $\varepsilon ^c$ is a basic evaluation. Conditions (i)–(v) follow immediately from the maximal consistency of $\varGamma$ and axioms of $\textsf{j}- \textsf{j5}$. Conditions (1)–(6) are obtained from the axioms (Abs), (COK), (Nec), (Id), (Ext), and (Sh). Let us only show (1) and (3).

To check condition (1), suppose $(C,B) \in \varepsilon ^c (t) \ominus \varepsilon ^c (s)$. Then there is an $A \in \textsf{Fm}$ such that $(A \rightarrow C , B) \in \varepsilon ^c (t)$ and $(A,B) \in \varepsilon ^c (s)$. By the definition of canonical model $[t](A\rightarrow C /B) \in \varGamma$ and $[s](A/B) \in \varGamma$, by maximal consistency of $\varGamma$ and axiom (COK) we have $[t \cdot s](C/B) \in \varGamma$, which gives $(C/B) \in \varepsilon ^c (t \cdot s)$.

To check condition (3), suppose $(A,B) \in \textsf{n}(\varepsilon ^c (\lambda ))$. Then $A \in \varepsilon ^c (\lambda )$, which means $\lambda :A \in \varGamma$. By maximal consistency of $\varGamma$ and axiom (Nec) we get $[\textsf{n}(\lambda )](A/B) \in \varGamma$. By the definition of canonical model we conclude $(A,B) \in \varepsilon ^c (\textsf{n}(\lambda ))$. Thus $\varepsilon ^c$ is a basic evaluation.

The truth lemma states:

$$\begin{aligned} F \in \varGamma \ \ \text {iff} \ \ \varepsilon ^c \Vdash F , \end{aligned}$$

which is established as usual by induction on the structure of F. In case $F= [t](A/B)$, we have $[t](A/B) \in \varGamma$ iff $(A,B) \in \varepsilon ^c (t)$ iff $\varepsilon ^c \Vdash [t](A/B)$.

Due to axiom $\textsf{jt}$, $\varepsilon ^c$ is factive by the following reasoning: if $\varepsilon ^c \Vdash \lambda : F$, we get by the truth lemma that $\lambda : F \in \varGamma$. By the maximal consistency of $\varGamma$ we have $F \in \varGamma$ which means $\varepsilon ^c \Vdash F$ by the truth lemma.    $\square$