Truth (Satisfaction)

Satisfaction

Let $\mathfrak{A}$ be an $\mathcal{L}$-structure. We define for every variable assignment $\sigma$, by induction on the complexity of a formula $\phi$, the relation $\mathfrak{A}\models \phi [\sigma ]$:

1. $\mathfrak{A}\models (t_1=t_2)[\sigma ]$ iff $t_1^{\mathfrak{A}}[\sigma ]=t_2^{\mathfrak{A}}[\sigma ]$,
2. $\mathfrak{A}\models r(t_1,\dots ,t_m)[\sigma ]$ iff $(t_1^{\mathfrak{A}}[\sigma ],\dots ,t_m^{\mathfrak{A}}[\sigma ])\in r^{\mathfrak{A}}$,
3. $\mathfrak{A}\models (\neg \phi )[\sigma ]$ iff $\mathfrak{A}/\models \phi [\sigma ]$,
4. $\mathfrak{A}\models ({\phi }_1\wedge {\phi }_2)[\sigma ]$ iff $\mathfrak{A}\models {\phi }_1[\sigma ]$ and $\mathfrak{A}\models {\phi }_2[\sigma ]$,
5. $\mathfrak{A}\models \forall x\phi [\sigma ]$ iff $\mathfrak{A}\models \phi [{\sigma }^{\prime }]$ for all ${\sigma }^{\prime }$ such that $\sigma (u)={\sigma }^{\prime }(u)$ whenever $u\ne x$.

Universal Validity

If $\phi =\phi (x_1,\dots ,x_n)$ is a formula, then by $\mathfrak{A}\models \phi$ we mean that $\mathfrak{A}\models \phi [\sigma ]$ for all assignments $\sigma :\mathcal{V}\to A$

Semantic Entailment

Suppose $\Sigma$ is a set of formulas (possibly infinite) and $\phi$ is a formula from a language $\mathcal{L}$. Then $\Sigma \models \phi$ if for every $\mathcal{L}$-structure, $\mathfrak{A}$, if for all $\psi \in \Sigma$, $\mathfrak{A}\models \psi ⟹\mathfrak{A}\models \phi$.

Soundness Theorem

If $\Sigma$ is a set of formulas and $\phi$ is a formula(all in the same language $\mathcal{L}$), then $\Sigma ⊢\phi$, implies $\Sigma \models \phi$

Inference Preserves Validity

The two rules of inference preserve validity in the following sense: (i)if $\mathfrak{A}\models \phi$ and $\mathfrak{A}\models \phi \to \psi$ then $\mathfrak{A}\models \psi$, (ii) if $\mathfrak{A}\models \phi$ then $\mathfrak{A}\models \forall v\phi$ .