Mathematics
Logical Consequence and the Semantic Theory of Proof
Quick fact
The semantic theory of proof shows that a proof's validity rests on the absence of any counterexample—not just the ones we can think of, but all imaginable ones.
Why this is interesting
You know a valid argument when you see one—but what makes it valid? Surprisingly, it's not about the facts but about all possible worlds.
Read the full explanation
Understanding Logical Consequence and the Semantic Theory of Proof
Think of logical consequence as a strict promise: if the premises are true, the conclusion must be true. The semantic theory of proof explains this using 'possible worlds'—imagine every conceivable situation. An argument is valid (premises logically force the conclusion) if there is no possible world where the premises are true but the conclusion is false. For example, from 'All humans are mortal' and 'Socrates is human', the conclusion 'Socrates is mortal' follows because you can't imagine a world where the first two are true and the third is false. This is different from just checking one real-world case—we check all logical possibilities. Truth tables help us visualize this for basic logic: each row is a mini-world. If no row has true premises and a false conclusion, the argument is valid.
A deeper explanation
The semantic theory of proof formalizes logical consequence (⊨) as a relation between statements and models. A model is a precise 'possible world' that assigns truth values to every atomic proposition. An argument is semantically valid if every model satisfying the premises also satisfies the conclusion. This contrasts with syntactic proof (⊢), where rules of inference generate conclusions step-by-step from premises. The central theorems—soundness and completeness—link these: soundness says if you can prove something, it is semantically valid; completeness says if something is semantically valid, you can prove it (in a sufficiently powerful system). This duality reveals why proofs are trustworthy: they capture all possible truths. The theory also extends to predicates with quantifiers, requiring models with objects and relations, not just truth assignments. This underpins all of mathematics: each theorem is a logical consequence of axioms in every model, ensuring absolute certainty within the formal system.