Question

Looking for examples that show me how to quantify over actions (and perhaps fluents?) in situation calculus (Reiter 2001).

I understand the difference between actions, fluents and situations, but why do they need to be represented in 2nd order logic? Why not use first order? Can you please explain?

Answer 1

All but a few formulas used to encode a dynamic world require Second Order Logic (SOL). In particular,

• the [properties of the] Initial State
• the Action Preconditions
• the Action Effects
• the Successor State axioms (used to circumvent the “Frame problem”)

can all be expressed with First Order Logic (FOL).
With some domains it may be convenient -and more concise- to use SOL for the above items, but AFAIK, it is always possible to convert such SOL to FOL in the context of the items listed above, for a finite domain, and hence SOL is not necessary (again, for the above items).

Typically the need for SOL with Situation Calculus only comes from some of the “Foundational axioms” such as the axiom to perform induction on Situations.

Furthermore, depending on the particular application, one may not need the SOL-based Foundational axiom(s) and hence the whole world can be exclusively described in terms of FOL.
I’m not a specialist in the field, but I think that in many cases, for the purpose of [Logical] Filtering, Planning, Temporal Projection, we can do away with the need for induction and hence only rely on FOL only.