This document outlines a formal system called metamathematics of contexts. It describes contexts as sets that define the meaningful vocabulary within that context. Formulas are built from propositional atoms using context operators. Models are functions that map context sequences to partial truth assignments. The semantics define satisfaction in a model based on the vocabulary of a context. The system has provability rules and useful theorems. Extensions examine consistency models where contexts cannot assign different truth values, truth models with a single assignment, and flatness models where contexts have identical vocabularies regardless of sequence.

1. METAMATHEMATICS OF CONTEXTS

2. Outline
Introduction to contexts
The general system
Notation
Syntax
Semantics
Models
Vocabularies
Satisfaction
Provability
Useful theorems
Extensions for the general system
Consistency Model
Truth Model
Flatness Model

3. How context formalism can be useful?
• In the context of situation calculus
• On(x , y , s) – Object x is on top of object y in situation s.
• Above(x , y , s) – Situation calculus does not have a definition for above.
So, using context formalism, the agent can (import) the definition of above
from the context of common sense knowledge.
E.g. Above means on. The agent then should relate that Above(x , y, s)
means On(x , y , s)

4. Propositional logic of contexts
• modality is used to express that sentence
holds in context
• Each context has it’s own vocabulary.
• The vocabulary of a context is the set of atoms that are
meaningful in that context.

5. Notation
• Given that X and Y are Sets then:
• is the set of partial functions from X to Y.
• is the set of subsets of X.
• is the set of all finite sequences in X that can be treated as a tree.
• is a range over .
• is the empty sequence.

6. Syntax
• Let be the set of all contexts, P be the set of all propositional
atoms.
• We can now build the set of all well-formed formulas (wffs)
using K and P in the following recursive fashion:
• We will also be using the following abbreviations

7. Semantics - Model
• In this system, a model , will be a function which maps a
context sequence to a set of partial truth assignments denoted
by or .
• Why a context sequence instead of a single context?
• The truth assignments need to be partial. Why?

8. Semantics - Vocabularies
• A Vocabulary of a context is the set of atoms that are
meaningful in that context.
• is a function that given a model , returns the
vocabulary for that model.
• Different contexts can have different vocabularies. That’s why
the truth assignments need to be partial.

9. Semantics - Satisfaction
•
Why? Because we add a third logic value other than true, false. So if X is not true, it
doesn’t have to be false.

10. Provability
• A formula is provable in context with vocabulary Vocab iff it is an instance of an
axiom schema or follows from provable formulas by the inference rules mentioned
above.

11. Useful Theorems
Ps. The previous theorems are proved using the axioms, inference rules in the
previous slide.

12. System Extensions - Consistency
• Sometimes it’s desirable to ensure that all contexts are
consistent.
• In this extension we examine the class of consistent models
. A model iff for any context sequence in
the domain of that model holds.
• In other words, if no two truth assignments give different truth
values for the same atom, then the model that maps the
context to these truth assignments can be described as
consistent.

13. System Extension - Truth
• A model is a truth model, formally iff for any
context sequence in the domain of that model,
• In other words, if the model has only one or less truth
assignment function, it can be described as a truth model.

14. System Extension - Flatness
• For some applications, all contexts will be identical regardless
of which context are they viewed from. This is called flatness.
• A model is flat, formally , iff for any context
sequences and any context

15. Questions 

16. Thank you