Simple situation calculus

We begin with some axioms in a formalism like that of [McC86] using a $Result$ function, a single abnormality predicate and aspects. This suffers from the Yale shooting problem if we simply minimize $Ab$. However, as long as the problem requires only projection, i.e. predicting the future from the present without allowing premises about the future, chronological minimization of $Ab$ [Sho88] avoids the Yale shooting problem. It is certainly a limitation on elaboration tolerance to not allow premises about the future.

Here are some axioms and associated issues of elaboration tolerance.

The basic operation of moving some people from one bank to the other is conveniently described without distinguishing between missionaries and cannibals.

$$\begin{array}[l]{l} \lnot Ab(Aspect1(group,b1,b2,s)) \right... ...quad\quad Value(Inhabitants(b2),s) \cup group, \\ \end{array}$$ (1)

where $\setminus$ denotes the difference of sets.

The fact that (1) can't be used to infer the result of moving a group if some member of the group is not at $b1$ is expressed by

$$\lnot (group \subset Value(Inhabitants(b1),s)) \rightarrow Ab(Aspect1(group,b1,b2,s)).$$

We extend the notion of an individual being at a bank to that of a group being at a bank.

$$Holds(At(group,b),s) \equiv (\forall x \in group)Holds(At(x,b),s).$$

$$\begin{array}[l]{l} \lnot Ab(Aspect2(group,b1,b2,s)) \land... ...d\quad \rightarrow \lnot Ab(Aspect1(group,b1,b2,s)) \end{array}$$ (2)

relates two abnormalities.

$$Crossable(group,b1,b2,s) \rightarrow 0 < Card(group) < 3$$ (3)

tells us that the boat can't cross alone and can't hold more than two.

$Card(u)$ denotes the cardinality of the set $u$.

We can sneak in Jesus by replaceing (3) by

$$Crossable(group,b1,b2,s) \rightarrow 0 < Card(group \setminus \{Jesus\}) < 3,$$ (4)

but this is not in the spirit of elaboration tolerance, because it isn't an added sentence but is accomplished by a precise modification of an existing sentence (3) and depends on knowing the form of (3). It's education by brain surgery.

It's bad if the cannibals outnumber the missionaries.

$$\begin{array}[c]{l} Holds(Bad(bank),s) \\ \equiv \\ 0 < ... ...\vert x \in Cannibals \land Holds(At(x, bank),s)\}) \end{array}$$ (5)

and

$$Holds(Bad,s) \equiv (\exists bank)Holds(Bad(bank),s).$$ (6)

Many unique names axioms will be required. We won't list them in this version.