Consequences of a Theory (First-Order Logic)

first-order-logic

Definition

Consequences of a Theory (First-Order Logic)

For a theory $\Gamma$:

$$\mathrm{Cons}(\Gamma )=\{\phi \mid \Gamma \models \phi \}$$

Here $\phi$ ranges over sentences. $\mathrm{Cons}(\Gamma )$ collects every sentence entailed by $\Gamma$.

Monotonicity

Theorem

Adding sentences grows the set of consequences:

$$\mathrm{Cons}(\Gamma )\subseteq \mathrm{Cons}(\Gamma \cup \{\phi \})$$