Metamathematics of contexts
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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.
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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 its 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. Thats 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 doesnt 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 its 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