Semantic theory of truth

Satisfaction

In the modern model theoretic formulation, suppose ${\displaystyle L}$ is a first-order language, with ${\displaystyle {\mathcal {M}}}$ an ${\displaystyle L}$-structure with a domain ${\displaystyle M}$, and with ${\displaystyle s}$ an assignment that maps variables to elements of ${\displaystyle M}$.

In general, ${\displaystyle {\mathcal {M}},s\models \varphi }$ indicates that the assignment ${\displaystyle s}$ satisfies the formula ${\displaystyle \varphi }$ in ${\displaystyle {\mathcal {M}}}$. Thereafter, satisfaction can be defined recursively on the construction of formulas.

• Atomic Formula: ${\displaystyle R(t_{1},\ldots ,t_{n})}$, ${\displaystyle {\mathcal {M}},s\models R(t_{1},\ldots ,t_{n})}$ iff. the values of ${\displaystyle t_{1},\ldots ,t_{n}}$ under ${\displaystyle {\mathcal {M}}}$ and ${\displaystyle s}$ form an n-tuple belonging to the relation interpreting R in ${\displaystyle {\mathcal {M}}}$.

For example, ${\displaystyle {\mathcal {M}},s\models \neg \varphi }$ if and only if ${\displaystyle {\mathcal {M}},s\not \models \varphi }$, and ${\displaystyle {\mathcal {M}},s\models \varphi \land \psi }$ if and only if ${\displaystyle {\mathcal {M}},s\models \varphi }$ and ${\displaystyle {\mathcal {M}},s\models \psi }$.

For quantifiers, vary the assignment to the quantified variable.

• For All: ${\displaystyle {\mathcal {M}},s\models \forall x\,\varphi }$ iff. ${\displaystyle {\mathcal {M}},s[x\mapsto a]\models \varphi }$ for every ${\displaystyle a\in M}$, where ${\displaystyle s[x\mapsto a]}$ denotes the assignment that agrees with ${\displaystyle s}$ on every variable except possibly x, to which it assigns a.
• Exists: ${\displaystyle {\mathcal {M}},s\models \exists x\,\varphi }$ if and only if ${\displaystyle {\mathcal {M}},s[x\mapsto a]\models \varphi }$ for at least one ${\displaystyle a\in M}$.

Truth for sentences

A formula with no free variables is defined as a sentence. Here, satisfaction is not dependent on the choice of assignment. A sentence ${\displaystyle \varphi }$ is said to be true in ${\displaystyle {\mathcal {M}}}$, denoted

${\displaystyle {\mathcal {M}}\models \varphi ,}$

if ${\displaystyle {\mathcal {M}},s\models \varphi }$ for every assignment ${\displaystyle s}$ (equivalently, some assignment ${\displaystyle s}$). Note that truth is a special case of satisfaction for sentences.