Satisfaction in Model Theory

Understanding Satisfaction in Model Theory

Satisfaction is a core concept in model theory, defining the truth of a sentence within a given structure. It establishes the fundamental link between formal languages and their interpretations.

Key Concepts

• Structure: A mathematical object that provides an interpretation for the symbols in a formal language.
• Sentence: A formula in a formal language with no free variables.
• Interpretation: Assigning meaning to the non-logical symbols of a language within a structure.

The Satisfaction Relation

A structure M satisfies a sentence φ (denoted M ⊨ φ) if and only if φ is true in M under the given interpretation of its symbols.

Deep Dive

Satisfaction is formally defined recursively. For atomic sentences, satisfaction depends on the interpretation of relation symbols. For sentences involving logical connectives (¬, ∧, ∨, →, ↔) and quantifiers (∀, ∃), the truth conditions are built upon the satisfaction of sub-formulas within the structure.

Applications

The notion of satisfaction is crucial for:

• Model checking in computer science.
• Proving independence results in mathematical logic.
• Developing theories of computation and computability.

Challenges and Misconceptions

A common misconception is confusing satisfaction with syntactic validity. A sentence can be satisfied by some structures but not others. Model existence is a related but distinct concept.

FAQs

Q: What is the difference between satisfaction and truth?
A: In this context, satisfaction is the formal definition of truth within a specific structure.

Q: Does every sentence have a structure that satisfies it?
A: Not necessarily. This depends on the properties of the sentence and the theory it belongs to.