Model-Theoretic Consequence

Overview

Model-theoretic consequence, often synonymous with semantic consequence, is a cornerstone of formal logic. It defines what it means for a conclusion to logically follow from a set of premises.

Key Concepts

The core idea is that a conclusion is a model-theoretic consequence of premises if and only if every interpretation (or model) that makes all the premises true also makes the conclusion true. This is a semantic notion, focusing on the meaning and truth conditions of statements.

Deep Dive

In formal logic, a set of sentences $\Gamma$ semantically entails a sentence $\phi$ (denoted $\Gamma \models \phi$) if, for every structure $\mathcal{M}$ (a model), if $\mathcal{M} \models \psi$ for all $\psi \in \Gamma$, then $\mathcal{M} \models \phi$. This contrasts with syntactic consequence, which is based on proof rules.

Applications

Model-theoretic consequence is vital in:

• Database theory: Ensuring query answers are correct based on data interpretations.
• Automated theorem proving: Guiding the search for proofs by preserving truth.
• Formal verification: Specifying and verifying system properties.

Challenges & Misconceptions

A common misconception is confusing it with syntactic consequence. While often equivalent in classical logic, they are distinct concepts. The complexity of defining and working with models can be a challenge.

FAQs

What is a model? A mathematical structure that interprets the symbols of a formal language. How does it relate to truth? It defines logical consequence based on truth preservation in all models.