Semantic Consequence

Semantic consequences occur when we can take a formula and are able to make some inferences that hold true whenever that formula itself is true (which can be done once it has been assigned meaning). Formally, a semantic consequence of a formula remains true for every model of it.

Semantic consequences is a type of meta-logic, that is, we use logic to talk about logic. It is important to note that such meta-logic exists outside any formal logic (like propositional logic). In other words, we can talk about the semantic consequences of any type of logic, not just propositional logic.

Definition §

Semantic Consequence ^definition

We say that a formula $G$ is a semantic consequence of a formula $F$ if, and only if, every model of $F$ is a model of $G$. This is written as:

$$F\models G$$

Alternatively, $F$ logically entails $G$

Theorems §

T1: Logical entailment equivalent to validity

Let $F$ and $G$ be any two formulae. Then

$$(F\models G)\text{ iff }[(\models F)⟹(\models G)]$$

In English: $F$ logically entails $G$ if, and only if, $F$ being valid implies $G$ is also valid. In this sense, if $F$ is valid, and $F\models G$, we know that $G$ is valid.

• $\models$: Logical entailment
• $⟹$: Universal or Meta-implication (not to be confused with material implication, which is only in propositional logic)

Entailment does not preserve invalidity! $F\models G$ and $F$ is non-valid or invalid, that does not imply that $G$ is also non-valid!

If we have

C1: In propositional logic, a valid material implication is equivalent to semantic consequences

Let $F$ and $G$ be two propositions (which are a type of formulae). Then:

$$(F\models G)\text{ iff }(\models (F\to G))$$

In English: F logically entails G (or G is a semantic consequence of F) if, and only if, every model of F is also a model of G. That is, $F\to G$ is valid.

• $\models$: Logical entailment
• $\to$: Material Implication

T2: Logical entailment equivalent to unsatisfiability

Let $F$ and $G$ be any two formulae. Then

$$(F\models G)\text{ iff }[(F\models \perp )⟹(G\models \perp )]$$

In English: $F$ logically entails $G$ if, and only if, $F$ being unsatisfiable implies $G$ is also unsatisfiable.

• $\models$: Logical entailment
• $⟹$: Universal or Meta-implication (not to be confused with material implication, which is only in propositional logic)

T3: Unsatisfiability Theorem ^t3

For any two formulae under some logic

$$(F\models G)\text{ iff }((F\wedge \neg G)\models \perp )$$

• The expression on the left hand side of the iff says that $G$ is a semantic consequence of $F$ (or $F$ logically entails $G$).
• The expression on the RHS of iff says that $F\wedge \neg G$ is unsatisfiable

Proof T1 and a theorem about validity:

This corollary can be obtained by using

\begin{align} (F \models G) &\iff \models(F \to G) \tag{T1} \\ \models (F\to G) &\iff \lnot(F \to G) \models \bot \tag{Validity T1} \\ \lnot ( F \to G) &\equiv \lnot (\lnot F \lor G) \tag{Implication Equivalence} \\ &\equiv (\lnot\lnot F \land \lnot G) \tag{De Morgan Equivalence} \\ &\equiv (F \land \lnot G) \tag{Double Negation Equivalence} \\ (F \models G) &\iff (F \land \lnot G) \models \bot

\end{align}

T4: Logical entailment preserves satisfiability

Let $F$ and $G$ be any two formulae, and $M$ be a model. Then

$$(F\models G)⟹[(M\models F)⟹(M\models G)]$$

In English: If we know that $F$ entails $G$, than if $F$ is satisfiable, we also know that $G$ is satisfiable (using the same model $M$ that satisfied $F$)

• $\models$: Logical entailment
• $⟹$: Universal or Meta-implication (not to be confused with material implication, which is only in propositional logic)

Resources §

• https://philosophy.stackexchange.com/questions/10785/semantic-vs-syntactic-consequence