# Notes - Logic and Proof MT24, Logical theories

> Source: https://ollybritton.com/notes/uni/part-b/mt24/logic-and-proof/notes/logical-theories/ · Updated: 2025-05-10 · Tags: uni, notes

- [Course - Logic and Proof MT24](https://ollybritton.com/notes/uni/part-b/mt24/logic-and-proof/)
	- [Notes - Logic and Proof MT24, First-order logic](https://ollybritton.com/notes/uni/part-b/mt24/logic-and-proof/notes/first-order-logic/)

### Flashcards
Suppose $\sigma$ is a signature. @Define a theory $\pmb T$.::

A set of $\sigma$-sentences that is closed under semantic entailment, i.e. if $\pmb T \models F$, then $F \in \pmb T$.

Suppose $\sigma$ is a signature and $\mathcal A$ is a $\sigma$-structure. @Define $\text{Th}(\mathcal A)$.::

The set of sentences that are satisfied in $\mathcal A$.

Suppose $\pmb S$ is a set of sentences. How can you construct a theory, and what name is sometimes given to $S$ in this context?

(recall a theory is a set of sentences closed under entailment, although some sources define a theory just as a set of sentences without the entailment condition).

::

$$
\pmb T = \{F : \pmb S \models F\}
$$
The elements of $\pmb S$ are sometimes called the axioms of $\pmb T$.

@Define what it means for a theory $\pmb T$ to be complete.::

For any sentence $F$, either $F \in \pmb T$ or $\lnot F \in \pmb T$.

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