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+# Basic terminology
+
+A **signature** (with relation names and arities) is defined [like in databases](Basic database terminology#signatures). A **structure** for a signature $\sigma$ consists of a **domain** of elements and an **interpretation** for each relation $R$ of $\sigma$, i.e., a set of $n$-tuples of elements of the domain, where $n$ is the arity of $R$. This definition is similar to that of a [database instance](Basic database terminology#instances) except that the domain is explicitly given, and it may be the case that some domain elements do not occur in the interpretation of any relation.
+
+# First-order logic
+
+A **first-order logic** formula is defined over a [signature](Basic database terminology#signatures) $\sigma$, and consists of an expression built from [atoms](Basic database terminology#facts-and-atoms) and using
+
+- the logical connectives $\land$ (AND), $\lor$ (OR), and $\neg$ (NOT), as well as connectives that can be defined using those, e.g., the implication $\rightarrow$ (where $x \rightarrow y$ is equivalent to $\not x \lor y$)
+- optionally, **equality atoms** of the form $x = y$ (see [Equality](#equality) below)
+- existential quantification $\exists x ~ \phi$
+- universal quantification $\forall x ~ \phi$
+
+## Bound and free variables
+
+TODO inductive def
+
+A formula is called **Boolean** if it has no free variables.
+
+## Constants
+
+A formula features **constants** if TODO.
+
+## Equality
+
+We may talk of first-order logic **with equality**, or **without equality**, to indicate whether formulas are allowed to contain
+
+## Satisfaction of a formula
+
+The notion of a structure $S$ **satisfying** a Boolean FO formula $\phi$ is defined inductively on the subformulas of $\phi$ in the expected way:
+
+TODO
+
+If $\phi$ holds in $S$, we can equivalently say that $S$ **satisfies** $\phi$.
+
+Satisfaction can also be defined using [Logical games]().
+
+## Examples
+
+One example of a first-order logic formula is the Boolean formula $\phi: \forall x ~ R(x) \rightarrow \exists y ~ S(x, y)$, stating that for every $x$ in the unary relation $R$ there exists an $y$ to which it is connected by the binary relation $S$. This formula $\phi$ is also a [tuple-generating dependency](Constraints#tuple-generating-dependencies).
+
+Another example is the formula $\psi: \forall x \, y \, y' ~ R(x, y) \land R(x, y') \rightarrow y = y'$. This formula $\psi$ is also a [functional dependency](Constraints#functional-dependencies)
+
+## Inexpressivity results and useful properties
+
+- Consider the signature $\sigma$ of [directed graphs](Basic graph terminology#directed-graph) consisting of one binary relation. It is known that first-order logic cannot express the formula $\phi$ asserting that the graph is connected: this is a consequence of [locality]().
+
+- The [zero-one law]() intuitively states that the probability of any first-order logic formula is either 0 or 1 as the domain size of structures goes to infinity.
+
+# Fixpoints
+
+TODO
+
+# Second-order logic
+
+TODO
+
+# Problems
+
+The [model-checking problem]() asks, given a Boolean logical formula $\phi$ and a structure $S$, whether $\phi$ holds in $S$. It is the analogue of the [query evaluation problem]() in databases.
+
+The [satisfiability problem]() asks, given a Boolean logical formula $\phi$, whether there is a structure $S$ that satisfies $\phi$.
