Hybrid-Logical Reasoning in the Smarties and Sally-Anne Tasks

Abstract

The main aim of the present paper is to use a proof system for hybrid modal logic to formalize what are called false-belief tasks in cognitive psychology, thereby investigating the interplay between cognition and logical reasoning about belief. We consider two different versions of the Smarties task, involving respectively a shift of perspective to another person and to another time. Our formalizations disclose that despite this difference, the two versions of the Smarties task have exactly the same underlying logical structure. We also consider the Sally-Anne task, having a more complicated logical structure, presupposing a “principle of inertia” saying that a belief is preserved over time, unless there is belief to the contrary.

Proof of Analyticity

Usually, when considering a natural deduction system, one wants to equip it with a normalizing set of reduction rules such that normal derivations satisfy the subformula property. Normalization says that any derivation by repeated applications of reduction rules can be rewritten to a derivation which is normal, that is, no reduction rules apply. From this it follows that the system under consideration is analytic.

Now, the works Braüner (2004) and Braüner (2011), subsection 4.3, by the present author devise a set of reduction rules for $\mathbf{N'}_{\mathcal{H}}$ obtained by translation of a set of reduction rules for a more common natural deduction system for hybrid logic. This more common system, which we denote $\mathbf{N}_{\mathcal{H}}$, can be found in Braüner (2004) and in Braüner (2011), subsection 2.2. All formulas in the system $\mathbf{N}_{\mathcal{H}}$ are satisfaction statements. Despite other desirable features, it is not known whether the reduction rules for $\mathbf{N'}_{\mathcal{H}}$ are normalizing, and normal derivations do not always satisfy the subformula property. In fact, Chapter 4 of the book Braüner (2011) ends somewhat pessimistically by exhibiting a normal derivation without the subformula property. It is remarked that a remedy would be to find a more complete set of reduction rules, but the counter-example does not give a clue how such a set of reduction rules should look.

In what follows we shall take another route. We prove a completeness result saying that any valid formula has a derivation in $\mathbf{N'}_{\mathcal{H}}$ satisfying a version of the subformula property. This is a sharpened version of a completeness result for $\mathbf{N'}_{\mathcal{H}}$ originally given in Braüner (2004) and in subsection 4.3 of Braüner (2011) (Theorem 4.1 in Braüner (2011)). Thus, we prove that $\mathbf{N'}_{\mathcal{H}}$ is analytic without going via a normalization result. So the proof of the completeness result does not involve reduction rules. The result is mathematically weaker than normalization together with the subformula property for normal derivations, but it nevertheless demonstrates analyticity. Analyticity is a major success criteria in proof-theory, one reason being that analytic provability is a step towards automated theorem proving (which obviously is related to Leibniz’ aim mentioned in the intoduction of the present paper).

In the proof below we shall refer to $\mathbf{N}_{\mathcal{H}}$ as well as a translation $( \cdot )^{\circ }$ from $\mathbf{N}_{\mathcal{H}}$ to $\mathbf{N'}_{\mathcal{H}}$ given in Braüner (2004) and subsection 4.3 of Braüner (2011). This translates a derivation $\pi$ in $\mathbf{N}_{\mathcal{H}}$ to a derivation $\pi ^{\circ }$ in $\mathbf{N'}_{\mathcal{H}}$ having the same end-formula and parcels of undischarged assumptions. The reader wanting to follow the details of our proof is advised to obtain a copy of the paper Braüner (2004) or the book Braüner (2011).

The translation $( \cdot )^{\circ }$ satisfies the following.

Lemma 1

Let $\pi$ be a derivation in $\mathbf{N}_{\mathcal{H}}$. Any formula $\theta$ occuring in $\pi ^{\circ }$ has at least one of the following properties.

• 1. $\theta$ occurs in $\pi$.

• 2. $@_{ a}\theta$ occurs in $\pi$ for some satisfaction operator $@_{ a}$.

• 3. $\theta$ is a nominal $a$ such that some formula $@_{ a}\psi$ occurs in $\pi$.

Proof

Induction on the structure of the derivation of $\pi$. Each case in the translation $( \cdot )^{\circ }$ is checked.

Note that in item 1 of the lemma above, the formula $\theta$ must be a satisfaction statement since only satisfaction statements occur in $\pi$. In what follows $@_{ d}\Gamma$ denotes the set of formulas $\{ @_{ d} \xi \; | \; \xi \in \Gamma \}$.

Theorem 1

Let $\pi$ be a normal derivation of $@_{ d}\phi$ from $@_{ d} \Gamma$ in $\mathbf{N}_{\mathcal{H}}$. Any formula $\theta$ occuring in $\pi ^{\circ }$ has at least one of the following properties.

• 1. $\theta$ is of the form $@_{ a}\psi$ such that $\psi$ is a subformula of $\phi$, some formula in $\Gamma$, or some formula of the form $c$ or $\lozenge c$.

• 2. $\theta$ is a subformula of $\phi$, some formula in $\Gamma$, or some formula of the form $c$ or $\lozenge c$.

• 3. $\theta$ is a nominal.

• 4. $\theta$ is of the form $@_{ a}(p \rightarrow \bot )$ or $p \rightarrow \bot$ where $p$ is a subformula of $\phi$ or some formula in $\Gamma$.

• 5. $\theta$ is of the form $@_{ a}\bot$ or $\bot$.

Proof

Follows from Lemma 1 above together with Theorem 2.4 (called the quasi-subformula property) in subsection 2.2.5 of Braüner (2011).

We are now ready to give our main result, which is a sharpened version of the completeness result given in Theorem 4.1 in subsection 4.3 of Braüner (2011).

Theorem 2

Let $\phi$ be a formula and $\Gamma$ a set of formulas. The first statement below implies the second statement.

• 1. For any model $\mathcal{M}$, any world $w$, and any assignment $g$, if, for any formula $\xi \in \Gamma$, $\mathcal{M},g,w \models \xi$, then $\mathcal{M},g,w \models \phi$.

• 2. There exists of derivation of $\phi$ from $\Gamma$ in $\mathbf{N'}_{\mathcal{H}}$ such that any formula $\theta$ occuring in the derivation has at least one of the five properties listed in Theorem 1.

Proof

Let $d$ be a new nominal. It follows that for any model $\mathcal{M}$ and any assignment $g$, if, for any formula $@_{ d} \xi \in @_{ d} \Gamma$, $\mathcal{M},g \models @_{ d}\xi$, then $\mathcal{M},g \models @_{ d}\phi$. By completeness of the system $\mathbf{N}_{\mathcal{H}}$, Theorem 2.2 in subsection 2.2.3 of the book Braüner (2011), there exists a derivation $\pi$ of $@_{ d}\phi$ from $@_{ d} \Gamma$ in $\mathbf{N}_{\mathcal{H}}$. By normalization, Theorem 2.3 in subsection 2.2.5 of the book, we can assume that $\pi$ is normal. We now apply the rules $(@I)$, $(@E)$, and $({ Name})$ to $\pi ^{\circ }$ obtaining a derivation of $\phi$ from $\Gamma$ in $\mathbf{N'}_{\mathcal{H}}$ satisfying at least one of the properties mentioned in Theorem 1.

Remark

If the formula occurrence $\theta$ mentioned in the theorem above is not of one of the forms covered by item 4 in Theorem 1, and does not have one of a finite number of very simple forms not involving propositional symbols, then either $\theta$ is a subformula of $\phi$ or some formula in $\Gamma$, or $\theta$ is of the form $@_{ a}\psi$ such that $\psi$ is a subformula of $\phi$ or some formula in $\Gamma$. This is the version of the subformula property we intended to prove.